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Simultaneous estimation of the parameters of the distributions of independent Poisson random variables Tsui, Kam-Wah 1978

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SIMULTANEOUS ESTIMATION OF THE PARAMETERS OF THE DISTRIBUTIONS OF INDEPENDENT POISSON RANDOM VARIABLES  by KAM-WAH JTSITI B.Sc,  The C h i n e s e U n i v e r s i t y o f Hong Kong, 1970 M . S c , U n i v e r s i t y o f Windsor, 1974  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department o f Mathematics and I n s t i t u t e of A p p l i e d  Mathematics and S t a t i s t i c s  We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1978 ^)  Kam-Wah T s u i , 1978  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I  further  of  this  fulfilment of  the U n i v e r s i t y of B r i t i s h  s h a l l make it  freely  available  for  agree t h a t p e r m i s s i o n f o r e x t e n s i v e  for scholarly by h i s  in p a r t i a l  the requirements  Columbia,  I agree  for  that  reference and study. copying o f  this  thesis  purposes may be granted by the Head of my Department or  representatives. thesis for  It  financial  i s understood that gain s h a l l  not  copying or  publication  be allowed without my  written permission.  Department of Mathematics and Institute of Applied Mathematics and Statistics The U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  June 30, 1978  ii  Supervisors:  Professor S. James Press and Professor James V. Zidek ABSTRACT  This work is devoted to simultaneously estimating the parameters of the distributions of several independent Poisson random variables.  In  particular, we explore the possibility of finding estimators of the Poisson parameters which have better performance than the maximum likelihood estimator (MLE).  We f i r s t approach the problem from a fre-  quentist point of view, employing a generally scaled loss function, called the k-normalized squared error loss function P L  K.  (A,A)  =  , Z - 1( A .1  1=1  where k i s a non-negative integer.  - A.) / 2  A  K  1  ,  The case k=0' i s the squared error  loss case, in which we propose a large class of estimators including those proposed by Peng [1975] as special cases.  Estimators pulling the MLE  towards a point other than zero as well as a point determined by the data itself are proposed, and i t i s shown that these estimators dominate the MLE uniformly. Under L^ with k >_ 1, we obtain a class of estimators dominating the MLE which includes the estimators proposed by Clevenson and Zidek [1975]. We next approach the problem from a Bayesian point, of view; a twostage prior distribution i s adopted and results for a large class of prior distributions are derived. Substantial savings in terms of mean squared error loss of the Bayes point estimators over the MLE are expected, especially when the Poisson parameters f a l l into a relatively narrow range.  iii  An empirical Bayes approach to the problem i s carried out along the line suggested by Clevenson and Zidek [1975].  Some results are obtained  which parallel those of Efron and Morris [1973], who work under the assumption that the random variables are normally distributed. We report the results of our computer simulation to quantitatively examine the performance of some of our proposed estimators.  In most  cases, the savings, under the appropriate loss functions, are an in-: . creasing function of the number of Poisson parameters.  The simulation  results indicate that our estimators are very promising. of the Bayes estimators depend on the choice of prior  The savings  hyperparameters,  and hence proper choice leads to substantial improvement over the MLE. Although most of the results in this work are derived under the assumption that only one observation i s taken from each Poisson d i s t r i bution, we extend some results to the case where possibly more than one observation is taken.  We conclude with suggestions for further work.  iv  TABLE OF CONTENTS Page SECTION 1. INTRODUCTION 1.1 Background  1  1.2 Outline  5  SECTION 2. NOTATION AND FUNDAMENTALS 2.1  Notation  10  2.2 Basic Lemmas and Theorems  11  SECTION 3. ESTIMATION UNDER SQUARED ERROR LOSS 3.1 Introduction  19  3.2 Notation  20  3.3 Shifting the MLE Towards k  23  3.4 Adaptive Estimators  36  SECTION 4. ESTIMATION UNDER K-NORMALIZED SQUARED ERROR LOSS 4.1 Introduction  41  4.2 Minimax Estimators  41  4.3 Better Estimators Under L, k  t  50  SECTION 5. BAYESIAN ANALYSIS 5.1 Introduction  57  5.2 Estimates of the Parameters  59 •  5.3 Marginal Posterior Density  70  5.4  76  Summary  V  SECTION 6.  EMPIRICAL BAYES ESTIMATION  6.1  Background  6.2  R e l a t i v e Savings Loss i n the P o i s s o n Case  79  6.3  The P l u s Rules  81  6.4  Bayes Rules w i t h Respect  6.5  Truncated  6.6  The R i s k F u n c t i o n of the E s t i m a t o r A*  87  6.7  S i m p l i f i c a t i o n of R(b,A*)  89  6.8  R i s k F u n c t i o n of A* as a F u n c t i o n of A  91  6.9  R i s k F u n c t i o n of A-'as a F u n c t i o n of A .  94  6.10  R i s k F u n c t i o n of the Clevenson-Zidek  95  6.11  Summary  SECTION 7.  .. •  78  to Other P r i o r s  Bayes Rules  81 83  Estimators  96  COMPUTER SIMULATION  7.1  Introduction  7.2  E s t i m a t o r s Under k-NSEL  100  7.3  Bayes E s t i m a t o r s  103  7.4  E s t i m a t o r s Under Squared E r r o r Loss  109  7.5  Comparison of the E s t i m a t o r s  112  SECTION 8.  98  EXTENSIONS  8.1  Motivation  113  8.2  E s t i m a t o r s Under G e n e r a l i z e d Squared E r r o r Loss  114  8.3  E s t i m a t o r s Under G e n e r a l i z e d k-NSEL  117  8.4  An A p p l i c a t i o n  120  SECTION 9.  BIBLIOGRAPHY  SUGGESTIONS FOR  FURTHER RESEARCH  122  123  VI LIST OF TABLES Page I.  Improvement Percentage o f A  over the MLE  Narrow Range f o r X/s II.  101  Improvement Percentage o f A  over the MLE  Wide Range f o r A / s III.  101  Improvement Percentage o f A  over the MLE  Narrow Range f o r A / s  102 A  IV.  l  Improvement Percentage o f A  over the MLE  Wide Range f o r A / s V. VI.  102  Improvement Percentage o f A  over the MLE  P o i s s o n - D i s t r i b u t e d a Cu=0, v=0) Narrow Range f o r A ' s  104  P o i s s o n - D i s t r i b u t e d a (u=0, v=0) E f f e c t o f y  107  ±  VII. VIII.  104  P o i s s o n - D i s t r i b u t e d a (u=0, v=0) Wide Range f o r A.'s  107  l  IX. X.  N e g a t i v e B i n o m i a l - D i s t r i b u t e d a (u=l, v=0, a^=3.0) G e o m e t r i c - D i s t r i b u t e d a (u=l, v=0, a^=1.0)  XI.  (0) Improvement Percentage o f A over the MLE  108 108  A  Narrow Range f o r A / s  110 * (0)  XII.  Improvement Percentage of A  over the MLE  Wide Range f o r A s  110  ?  XIII.  Improvement Percentage o f A^ ^ over the MLE m  Narrow Range f o r A^'s XIV.  Improvement Percentage o f A Wide Range f o r A 's  I l l over t h e MLE I l l  vii:  ACKNOWLEDGEMENTS  I would l i k e to express my g r a t i t u d e to P r o f e s s o r James Zidek f o r s u g g e s t i n g my t h e s i s t o p i c and g u i d i n g my r e s e a r c h d u r i n g t h e f i r s t of my work. like  H i s keen i n s i g h t s were extremely  t h e t o p i c s s t u d i e d i n S e c t i o n s 4 and  Both P r o f e s s o r Zidek and P r o f e s s o r P r e s s p r o v i d e d encouragement and  s t i m u l a t i o n d u r i n g the p r e p a r a t i o n o f t h i s t h e s i s . extended to P r o f e s s o r S t a n l e y Nash f o r m e t i c u l o u s and  I would a l s o  t o thank P r o f e s s o r S. James P r e s s f o r s u p e r v i s i n g my work f o r the  p a s t two y e a r s , and f o r s u g g e s t i n g 5.  helpful.  year  Thanks a r e f u r t h e r reading of the t h e s i s ,  to P r o f e s s o r F r e d Wan, who has been v e r y h e l p f u l d u r i n g my e n t i r e ,  enrollment The  a t the U n i v e r s i t y of B r i t i s h  Columbia.  f i n a n c i a l a s s i s t a n c e o f the U n i v e r s i t y o f B r i t i s h Columbia and  the N a t i o n a l Research C o u n c i l o f Canada, as w e l l as t h e use o f t h e com- . p u t e r f a c i l i t i e s a t t h e U n i v e r s i t y o f B r i t i s h Columbia and the U n i v e r s i t y of C a l i f o r n i a , R i v e r s i d e , a r e g r a t e f u l l y acknowledged. F i n a l l y , I would l i k e t o thank my w i f e f o r h e r s t e a d f a s t encouragement and p a t i e n t a s s i s t a n c e .  1  SECTION 1. 1.1  INTRODUCTION  Background The l i t e r a t u r e on the problem o f e s t i m a t i n g the mean o f a p-dimen-  s i o n a l m u l t i v a r i a t e normal d i s t r i b u t i o n N ( 0 , E ) , b o t h when the c o v a r i a n c e P m a t r i x E i s known and unknown, has p r o l i f e r a t e d s i n c e S t e i n covered the s u r p r i s i n g r e s u l t is  dis-  t h a t when £ - I , the sample mean, which  the maximum l i k e l i h o o d e s t i m a t o r  e r r o r l o s s when p > 3.  [1956]  CMLE), i s i n a d m i s s i b l e under squared  B e t t e r e s t i m a t o r s o f the m u l t i v a r i a t e  normal  mean have been found by Alam [1973, 1975], B a r a n c h i k [1964, 1970], Berger [1976a, 1976b, 1976c], Berger and Bock [1976a, 1976b], Berger e t a l . [1977], E f r o n and M o r r i s James and S t e i n  [1973, 1976], H a f f [1976, 1977], Hudson [1974],  [1961], L i n and T s a i  [1971, 1973], and o t h e r s .  [1973], S t e i n  [1974], Strawderman  B a s i c a l l y , these e s t i m a t o r s a r e o b t a i n e d by  s h r i n k i n g the MLE towards a known f i x e d p o i n t or a p o i n t determined by the d a t a i t s e l f .  When we c o n s i d e r the problem from another p o i n t o f  view, we see t h a t f o r simultaneous e s t i m a t i n g o f s e v e r a l means o f normal p o p u l a t i o n s , i t i s i n a d m i s s i b l e to e s t i m a t e each p o p u l a t i o n mean by i t s sample mean, even though the p o p u l a t i o n s have no mathematical dependence and the o b s e r v a t i o n s between p o p u l a t i o n s a r e independent. A t t e n t i o n has a l s o r e c e n t l y been brought to simultaneous e s t i m a t i o n of the parameters o f non-normal distributions.  Johnson  d i s t r i b u t i o n s , i n c l u d i n g the d i s c r e t e  [1971] shows t h a t i n the b i n o m i a l case the u s u a l  e s t i m a t o r i s a d m i s s i b l e under squared e r r o r l o s s .  T h i s i s due to the  superb performance of the u s u a l e s t i m a t o r near the b o u n d a r i e s o f the p a r a meter space.  A l s o , attempts have been made t o work w i t h g e n e r a l members  of the e x p o n e n t i a l f a m i l y i n the simultaneous e s t i m a t i o n problem [1974, 1977]).  (Hudson  2  In t h i s study, we  c o n c e n t r a t e on the s i t u a t i o n where the u n d e r l y i n g  d i s t r i b u t i o n i s Poisson.  There a r e many p r a c t i c a l s i t u a t i o n s t h a t l e a d  us to i n v e s t i g a t e the problem of simultaneous meters of s e v e r a l independent  e s t i m a t i o n o f the p a r a -  Poisson d i s t r i b u t i o n s .  F o r example, a metro-  p o l i t a n area divided  i n t o s e v e r a l f i r e p r e c i n c t s might be  knowing the expected  numbers o f f i r e s  interested i n  i n each of the p r e c i n c t s i n a f i x e d  time p e r i o d so t h a t f i r e - f i g h t i n g r e s o u r c e s can be o p t i m a l l y a l l o c a t e d . During  t h a t p e r i o d , the number o f f i r e s  i n a s i n g l e p r e c i n c t c o u l d be  supposed to f o l l o w a P o i s s o n d i s t r i b u t i o n , w i t h an i n t e n s i t y parameter which would be d i f f e r e n t a c r o s s p r e c i n c t s . In the s t u d y of the p r o c e s s of o i l w e l l d i s c o v e r y by w i l d c a t e x p l o r a t i o n i n A l b e r t a , Canada, Clevenson the problem be  t r e a t e d as one  and  Zidek  that  of s i m u l t a n e o u s l y e s t i m a t i n g the parameters  of the d i s t r i b u t i o n s of s e v e r a l independent  P o i s s o n random v a r i a b l e s . ;  They l e t each parameter r e p r e s e n t the expected c o v e r i e s d u r i n g a p a r t i c u l a r month. o f parameters i s a p p r o x i m a t e l y  [1975] suggest  number o f / o i l w e l l  dis-  With t h e i r a v a i l a b l e d a t a , the number  200.  L e t x^,...,Xp be o b s e r v a t i o n s o f p independent'.Poisson random variables_X^,.. .yXp bwlth i n t e n s i t y parameters A^,..........A , r e s p e c t i v e l y and p >_ 2. Let x =  (x^,...,Xp), X =  (X^,.. .,-Xp), and  A =  (A^,..., A }.  to s i m u l t a n e o u s l y e s t i m a t e the parameters A^,...,A The u s u a l e s t i m a t o r of A i s the MLE,  X.  One  based on the d a t a  x.  might c o n j e c t u r e t h a t , as i n  the normal case, e s t i m a t o r s u n i f o r m l y b e t t e r than the MLE the P o i s s o n case when p i s l a r g e .  The problem i s  U s i n g the n o r m a l i z e d  can be found  squared  error  in loss  R "2 £ (A. - A.) /A., Clevenson and Z i d e k [1975] do indeed i=l o b t a i n a l a r g e c l a s s o f e s t i m a t o r s dominating the MLE as l o n g as p > 2. function  ^ L(A,A) =  3  Such a l o s s f u n c t i o n was chosen p a r t l y because t h e i r A^'s were expected to be s m a l l and hence i n a c c u r a t e e s t i m a t i o n o f s m a l l X_/s seemed h i g h l y u n d e s i r a b l e ; t h e i r l o s s f u n c t i o n r r e f l e c t s the d e s i r e t o p e n a l i z e  over-  e s t i m a t i o n o f s m a l l A.'s. The l o s s f u n c t i o n was chosen a l s o because the  x u s u a l e s t i m a t o r would be minimax i n t h i s case and because A. i s the i  1  population's variance.  Their estimators  s h r i n k the MLE towards t h e o r i g i n .  They g i v e two reasons why one might expect t h a t s h r i n k i n g the u s u a l estimator w i l l y i e l d a better  .';  estimator:  F i r s t l y the l o s s f u n c t i o n p e n a l i z e s h e a v i l y f o r bad e s t i m a t e s when the A.'s a r e s m a l l , and i n such cases i t i s o n l y p o s s i b l e t o produce bad o v e r e s t i m a t e s . Secondly S t e i n ' s r e s u l t s suggest i t i s b e t t e r to r e s t r a i n random m u l t i v a r i a t e and hence c h a o t i c o b s e r v a t i o n s by s h r i n k i n g them toward some p o i n t , here zero which does p l a y a d i s t i n g u i s h e d r o l e i n view o f the f i r s t r e a s o n above and because o f i t s s p e c i a l n a t u r e as the extreme p o i n t o f t h e parameter space. The  e s t i m a t i o n problem under d i f f e r e n t e r r o r l o s s f u n c t i o n s has been  s t u d i e d by o t h e r s .  Using  the squared e r r o r l o s s f u n c t i o n L ( A, X) =  P - 2 7. (A.-A.),, i=l 1  1  Hiidson [1974] proposes some e m p i r i c a l Bayes e s t i m a t o r s f o r A. The e s t i m a t o r s . _ a r e expected t o improve c o n s i d e r a b l y on X f o r a wide range o f parameter v a l u e s , e s p e c i a l l y when X^,...,A large.  a r e s i m i l a r i n v a l u e and a r e n o t too  But when one parameter A^ i s v e r y l a r g e and the.o.ther  s m a l l , the e s t i m a t o r s a r e i n f e r i o r expected t o be s m a l l . the MLE u n i f o r m l y .  parameters  t o the MLE X, though the l o s s i s  That i s , the proposed e s t i m a t o r s do n o t dominate  Hudson a l s o d e r i v e s an unbiased  improvement o f the r i s k when the e s t i m a t o r  e s t i m a t e U f o r the  A i s used i n s t e a d o f X, i . e .  E, U(X) = R(A,X) - R(A,A). A A s u f f i c i e n t c o n d i t i o n f o r A t o be u n i f o r m l y b e t t e r than X i s t h a t U(x) > 0 f o r a l l x and U(x) > 0 f o r some x.  Estimators  o f the form  4  A =  (11-  C(X))X, a r e of c o n s i d e r a b l e a p p e a l .  those o b t a i n e d  i n the normal case.  F o r , they a r e s i m i l a r  However, Hudson [1974] shows t h a t i n  order f o r the e s t i m a t o r s of t h i s form to y i e l d U(x) necessary  t h a t C(x)  >_ 0 f o r a l l x,  c l o s e r to the o r i g i n .  the problem i s then e s s e n t i a l l y one, be a d m i s s i b l e under squared  are z e r o , none of the  observations  T h i s i s because the dimension of  and when p = 1,^'the' M L E i i s known to  error loss  (Hodges and Lehmann  [1951]).  Because of the d i f f i c u l t y of h a n d l i n g cases when a l l but one parameters a r e near z e r o , i t was to improve upon the MLE  when squared  althoughtth'ezusual e s t i m a t o r when p >_ 3 by p r o p o s i n g  o r i g i n provided  One  of  c o n j e c t u r e d t h a t i t might be  the  impossible  e r r o r l o s s i s the c r i t e r i o n .  r e s u l t s of Peng [1975] a r e t h e r e f o r e q u i t e s u r p r i s i n g .  one u n i f o r m l y i n A.  i t is  =0.  When a l l the o b s e r v a t i o n s but one can be moved any  to  He  The  shows t h a t  i s a d m i s s i b l e when p ^_ 2, i t i s i n a d m i s s i b l e  e s t i m a t o r s t h a t a r e a c t u a l l y s u p e r i o r to the u s u a l of h i s e s t i m a t o r s s h r i n k s the MLE  towards the  the number of non^-zero o b s e r v a t i o n s i s g r e a t e r than two.  a l s o proposes an e s t i m a t o r which s h r i n k s a l l the non-zero components of MLE  towards z e r o , but which sometimes g i v e s non-^zero v a l u e s f o r the  components o f the MLE. of non-zero o b s e r v a t i o n s  The  the  zero  same c o n d i t i o n i s r e q u i r e d , namely the number  i s a t l e a s t t h r e e , i n o r d e r t h a t the proposed  e s t i m a t o r be s t r i c t l y b e t t e r than the  MLE.  So f a r , the e s t i m a t o r s t h a t a r e u n i f o r m l y b e t t e r than the MLE c e r t a i n l o s s f u n c t i o n s s h r i n k the MLE  towards the o r i g i n o n l y .  no o t h e r p o i n t towards which the s h r i n k a g e e s t i m a t o r s which s h r i n k the MLE i n t e g e r k, and  He  i s made.  We  under  There i s  propose some  towards an a r b i t r a r y prechosen nonnegative  some e s t i m a t o r s which s h r i n k the MLE  towards a p o i n t  5  determined by the d a t a . under squared  1.2  These e s t i m a t o r s dominate the u s u a l one u n i f o r m l y  error loss.  Outline In S e c t i o n 2, we i n t r o d u c e t h e n o t a t i o n t o be used i n the subsequent  s e c t i o n s and g i v e n some b a s i c lemmas t h a t a r e u s e f u l .  E s s e n t i a l l y , the  b a s i c lemmas (Lemmas 2.2.2 and 2.2.7) p r o v i d e i d e n t i t i e s o f the " r i s k d e t e r i o r a t i o n " under c e r t a i n l o s s f u n c t i o n s when an e s t i m a t o r X i s used i n s t e a d o f the MLE X. one  Two b a s i c theorems w i l l a l s o be s t a t e d .  (Theorem 2.2.3), due t o Peng 11975], p r o v i d e s e s t i m a t o r s o f t h e P o i s s o n  parameters which dominate the MLE u n i f o r m l y under squared The  The f i r s t  other  (Theorem 2.2.9), due t o Clevenson and Zidek  error loss.  [1975],  e s t i m a t o r s which dominate t h e MLE u n i f o r m l y under n o r m a l i z e d error loss.  An a l t e r n a t e p r o o f f o r the l a t t e r  provides squared  theorem i s p r o v i d e d .  S e c t i o n 3 examines the e s t i m a t i o n problem when the f a m i l i a r e r r o r l o s s f u n c t i o n i s used. and  Although  f i n d s e s t i m a t o r s dominating  e s t i m a t o r s i s expected  Peng [1975J  squared  i n v e s t i g a t e s the problem  the MLE when p >_ 3, the performance o f h i s  t o be good o n l y when a l l the u n d e r l y i n g parameters  a r e r e l a t i v e l y s m a l l o r when o n l y some o f the.parameters a r e l a r g e .  It  i s t h e r e f o r e o f i n t e r e s t t o see i f f u r t h e r improvements a r e p o s s i b l e .  In  "(k) t h i s s e c t i o n , we g e n e r a l i z e Peng's r e s u l t s and propose e s t i m a t o r s X which dominate t h e MLE u n i f o r m l y In- X:: k,  p u l l s the MLE towards k whenever the :he number o f o b s e r v a t i o n s  than k i s a t l e a s t t h r e e .  X)  F o r a f i x e d non-negative i n t e g e r  '=  X. + f . ( X ) , where  The e s t i m a t o r X  (k)  greater  = (X^,...,X ) i s g i v e n by  6  r <Kx)h(x ) k  f  1  ±  (x) = - —  if  -  E :.h j=i -  and r  , i = 1,.. . -, p  *y  (x.) J  i s the maximum of the number of o b s e r v a t i o n s  g r e a t e r than k,  less  K.  two,  and  3.2.  zero.  c o n d i t i o n s on t)>(x) and h(x^)  The  When the number of o b s e r v a t i o n s  are given i n S e c t i o n  g r e a t e r than k i s l e s s than  * (k)  three,  " (k)  A  g i v e s the same e s t i m a t e as the MLE.  When k=0,  X  reduces to  one  of Peng's e s t i m a t o r s . We  a l s o propose e s t i m a t o r s X^ m^  determined by the MLE  the d a t a i t s e l f .  which s h i f t  the MLE  towards a p o i n t  These a d a p t i v e e s t i m a t o r s , which dominate  u n i f o r m l y , a r e expected  to perform w e l l f o r a wide range of  parameter v a l u e s , i n c l u d i n g the case when the parameters, a r e a l l r e l a t i v e l y Fml  l a r g e but s i m i l a r i n v a l u e . A  X  The  estimator  A  i s s i m i l a r i n form to  (k) w i t h the f u n c t i o n s h ( x . ) , <J>(x)» and r, r e p l a c e d by a p p r o p r i a t e ones.  X  In S e c t i o n 4, we d i f f e r e n t angle.  K.  examine the simultaneous  I n a d d i t i o n to n o r m a l i z e d  e s t i m a t i o n problem from a  squared  l o s s , we  employ a more  g e n e r a l l o s s f u n c t i o n , namely, k-normalized square e r r o r l o s s (k-NSEL) 2 k L, (A,A) = E (A. - A.) /A. , where k i s a p o s i t i v e i n t e g e r . The case A  K  A  '  , _ X X X  1=1 when k=l, L^,  i s simply  which Clevenson and  the n o r m a l i z e d  Zidek  squared  [1975] perform  The u s u a l e s t i m a t o r X of  e r r o r l o s s f u n c t i o n under  their analysis.  A, under L^,  i s minimax.  Using  this  fact,  a s u f f i c i e n t c o n d i t i o n t h a t an e s t i m a t o r A of A be minimax under L^ i s t h a t i t s r i s k i s l e s s than or e q u a l to t h a t of X u n i f o r m l y i n A. 2.2.9  Theorem  thus p r o v i d e s us w i t h a c l a s s of minimax e s t i m a t o r s of A.  member of t h i s c l a s s i s of the form A(X) (|>(z) i s a nondecreasing  A  typical  = (1 - <j> (Z) / (Z+p-1) )X where  r e a l - v a l u e d f u n c t i o n and 0 <^  <)>(z)  2(p-l).  7  I n the f i r s t p a r t of t h i s s e c t i o n , we max  s h a l l show t h a t t h i s c l a s s of m i n i -  e s t i m a t o r s can be e n l a r g e d i n two ways.  F i r s t , we  include estimators  of the form  where (1) ip'(z) ^ b > 0 f o r some b (2)  (j)(z)  i s nondecreasing  for a l l (3) We  iKz)  z >_  and 0  £ <j>(z+l) £. 2Min{p-l, i|^(z) }  0  i s n o n i n c r e a s i n g and  z + iR.z)  is  nondecreasing.  next i n c l u d e e s t i m a t o r s of the form  x(x) = :(i  ^7Z?-  (z+b) where (1) t > 0, b > (2) <j>(z) i s t  )X  t + 1  t+1  nondecreasing  (3) <j>(z) >_ 0 and <|> (z) i 0 t  (4) «> (z) / ( z + b ) £ M i n { 2 (p-t-1) , 2 (b-t-1) }, t  Moreover, when b = p-1, <  )> (z)/(z+p-l)  t  t  c o n d i t i o n (4) can be r e p l a c e d by  < 2(p-t-1).  In the remainder of the s e c t i o n , we p r o v i d e m o t i v a t i o n f o r the A  of k-NSEL, L, , (k >_ 2) and d e r i v e e s t i m a t o r s X'~ the MLE  under L^.  A  (X  A  X )  B a s i c a l l y , the e s t i m a t o r s have the form  (KZ)X.(X.-l) •••(X.-k+l) X  l  = X. -  i  p  1  1  use  1  E (X.+l)(X.+2) •••(X..+k) + X . ( X . - l ) •••(X.-k+l) 2~j_ 3 3 3  dominating  8  where (1) 0 < cb(z) <_ 2 k ( p - l ) (2) (j)(z) i s n o n d e c r e a s i n g i n z. Because the MLE X i s a Bayes e s t i m a t o r i f p r i o r knowledge o f t h e i n t e n s i t y parameters  i s vague and t h e parameters a r e independent, i t i s  hoped t h a t s u b s t a n t i a l p r i o r knowledge, when i t i s a v a i l a b l e , can be i n c o r p o r a t e d i n a B a y e s i a n manner t o o b t a i n s i g n i f i c a n t l y b e t t e r e s t i m a t o r s of A than X.  I n S e c t i o n 5, we take a B a y e s i a n approach t o the problem  of s i m u l t a n e o u s l y e s t i m a t i n g t h e p i n t e n s i t y parameters when mean squared e r r o r i s the l o s s c r i t e r i o n .  The i n t e n s i t y parameters  changeable i n t h e sense o f de F i n e t t i  a r e assumed ex-  [1964], and t o j o i n t l y f o l l o w a  two-parameter gamma d i s t r i b u t i o n , a p r i o r i .  F o r t h e second s t a g e o f the  p r i o r d i s t r i b u t i o n we adopt a vague p r i o r d e n s i t y f o r one o f t h e gamma d i s t r i b u t i o n parameters, and a g e n e r a l i z e d hypergeometric d i s t r i b u t i o n f o r the o t h e r .  The j o i n t and m a r g i n a l p o s t e r i o r d e n s i t i e s o f the P o i s s o n  i n t e n s i t y parameters  a r e developed, and B a y e s i a n p o i n t e s t i m a t o r s a r e found.  In S e c t i o n 6, we f o c u s on e s t i m a t o r s o f X o f t h e form X = ( l - - " b ( Z ) ) X and use e m p i r i c a l Bayes methods t o perform a n a l y s i s a l o n g t h e l i n e by Clevenson and Zidek [1975].  suggested  A l t h o u g h no new e s t i m a t o r s a r e found i n  t h i s s e c t i o n , t h e approach w i l l be h e l p f u l i n g a i n i n g i n s i g h t s about some of t h e e s t i m a t o r s .  We c a l c u l a t e t h e " r e l a t i v e s a v i n g s l o s s " i n the P o i s s o n  case under n o r m a l i z e d square e r r o r l o s s and use i t as a t o o l to o b t a i n " p l u s rules"  (A  = (1 - b ( Z ) ) X ) and " t r u n c a t e d Bayes r u l e s " . +  We a l s o c a l -  c u l a t e t h e r i s k R(A,A ) o f t h e Clevenson-Zidek e s t i m a t o r s  V  5  = [1 - ( s ( p - l ) / (Z+p-1) ) ]X  9  's as a f u n c t i o n of X, where 0 <_ s <_ 2. We f i n d t h a t R(X,X ) , which i s P a c t u a l l y a f u n c t i o n of A = Z X., i s i n c r e a s i n g and concave i n A. s  i=l  1  S e c t i o n 7 c o n t a i n s the r e s u l t s of a computer s i m u l a t i o n used to q u a n t i t a t i v e l y compare the MLE those of Clevenson and  Zidek  w i t h some of our e s t i m a t o r s , as w e l l as [1975] and Peng [1975].  For each  estimator  X, the percentage of the s a v i n g s i n r i s k u s i n g X i n s t e a d of the MLE  is  A.  c a l c u l a t e d as  [^(^» ) x  R(X,X)-  >  2.00]%, u s i n g the l o s s f u n c t i o n under  K\A,Ji.)  which X was  derived.  In most of the c a s e s , the improvement p e r c e n t a g e i s  seen to be an i n c r e a s i n g f u n c t i o n of p, the number of d i s t r i b u t i o n s of the independent P o i s s o n random v a r i a b l e s .  For the non-Bayes e s t i m a t o r s ,  the improvement p e r c e n t a g e g e n e r a l l y decreases X^'s  increases.  The  as the magnitude of  improvement percentage of the Bayes e s t i m a t o r s depends  on the c h o i c e of p r i o r hyperparameters, and hence proper s u b s t a n t i a l improvement over In S e c t i o n 8, we MLE  the  the  c h o i c e l e a d s to  MLE.  r e t u r n to the s e a r c h f o r b e t t e r e s t i m a t o r s  than  under v a r i o u s l o s s f u n c t i o n s , but a l t e r the s e t t i n g by a l l o w i n g  f e r e n t numbers of o b s e r v a t i o n s  to be taken from each p o p u l a t i o n .  We  the  difare  l e d to c o n s i d e r weighted l o s s f u n c t i o n s , an example of which i s the P - 2 k g e n e r a l i z e d k-NSEL f u n c t i o n L ( X , X ) . - -2 c.(X. - X.) /X. . More i=l :  e s t i m a t o r s dominating the MLE  u n i f o r m l y i n X a r e proposed, and  c a t i o n to e s t i m a t i o n of the parameters of p independent P o i s s o n is  an  appli-  processes  provided. Finally,  S e c t i o n 9 c o n s i s t s of p r o p o s a l s  for further research.  10  SECTION 2.  NOTATION AND  FUNDAMENTALS  L e t X.,...,X be p independent P o i s s o n random v a r i a b l e s w i t h i n t e n s i t y 1 P parameters A^,...,A introduce  h e r e some n o t a t i o n  also include ensuing 2.1  , respectively  (p >_ 2 ) .  F o r easy r e f e r e n c e ,  we  t h a t w i l l be used throughout t h i s paper.  We  some b a s i c lemmas and theorems which prove t o be u s e f u l i n  sections.  Notation  Definitions  (1)  "(Random v a r i a b l e name) 'V ( D i s t r i b u t i o n name)" i n d i c a t e s t h a t the random v a r i a b l e has the s p e c i f i e d d i s t r i b u t i o n w i t h g i v e n p a r a m e t e r ( s ) .  (2)  X = (X^,...,X ); x = (x^,...,x ) i s the v e c t o r A= (A.,,...,A ) ; A = 1 P  P E i  =  1  A.. i  (3)  J = the s e t o f a l l i n t e g e r s ;  (4)  f.: J  p  J  +  = the s e t o f a l l n o n n e g a t i v e  -»• R, i = l , . . . , p , a r e f u n c t i o n s  p r o d u c t J* o f J w i t h i t s e l f 3  (5)  o f o b s e r v a t i o n s o f X;  f(X) = (f..(X),...,f (X)). X p  from the p - f o l d  integers.  Cartesian  t o t h e s e t o f r e a l numbers R.  E j f . ( X ) | < », i = l , . . . , p . A X  (k) (.6)  Y  = Y ( Y - l ) ... (Y-k+1)[, where k i s a p o s i t i v e i n t e g e r  real  number.  (7)  Z =  Z  (8) (9)  1=1 i=l <()(z) i s a nonr-decreasing r e a l r v a l u e d [ S ] = the w power o f S.  P  X.; z = . , x  W  th  P E x. . . , i function.  and Y i s a  11  til  (10)  e. = the p - v e c t o r whose i x  c o o r d i n a t e i s one and the r e s t o f  whose c o o r d i n a t e s a r e z e r o .  (11)  R(A,A) = t h e r i s k o f t h e e s t i m a t o r A o f A.  (12)  NB(u,p) = the n e g a t i v e b i n o m i a l d i s t r i b u t i o n w i t h mass f u n c t i o n P r ( z | b ) =  (13)  H  P = Max {x.}; i=l  N.  Z  z e J  +  .  1  = # {x. : x. = j'}. x  3  x  J  (15)  N = (N ,...,N ) . O. x.  (16)  (y)  2.2  P  P M i n {x.}. i=l  m=  1  (14)  z+p-1 ^ b (l^b) , z  probability-  +  = Max { y 0 } . 5  B a s i c Lemmas and Theorems Let  A be an e s t i m a t o r o f A and R(A,A) be t h e r i s k o f A.  estimators considered  Most o f our  i n subsequent s e c t i o n s w i l l be o f t h e form X + f ( X )  where f i s as d e f i n e d i n s u b s e c t i o n 2.1. D e f i n e t h e r i s k improvement o f A over  the MLE, X, to be  I = R(A,X) T, RCA,A). Hudson below.  [1974J f u r n i s h e s a u s e f u l i d e n t i t y which we s t a t e as Lemma  2.2.1  Based on t h a t i d e n t i t y , Hudson d e r i v e s a b a s i c i d e n t i t y about  the unbiased  r i s k improvement e s t i m a t e under squared e r r o r l o s s :  where U i s a f u n c t i o n o f X o n l y .  Peng  I E^ =  U(X),  11975] d e f i n e s r i s k d e t e r i o r a t i o n  D = - I = E (.^U(X))' and shows t h a t h i s e s t i m a t o r s s a t i s f y t h e i n e q u a l i t y A ^U(x)  <^ 0 f o r a l l x w i t h s t r i c t i n e q u a l i t y f o r some x.  This implies that  the proposed e s t i m a t o r s dominate the u s u a l one under squared The  error loss.  i d e n t i t y o f the r i s k d e t e r i o r a t i o n w i l l be s t a t e d as Lemma 2.2.2 below.  12  Use  of t h e same i d e n t i t y l e a d s t o t h e d i s c o v e r y of s t i l l more e s t i m a t o r s of X  better  than the MLE under squared e r r o r l o s s .  We s h a l l show i n S e c t i o n  3 t h a t t o any f i x e d nonnegative i n t e g e r k, t h e r e corresponds a c l a s s o f estimators  o f X which s h r i n k s the MLE towards k as l o n g as the number o f  v a r i a b l e s i s a t l e a s t ;three.  These e s t i m a t o r s w i l l  g i v e an. e s t i m a t e  d i f f e r e n t from the MLE i f the number o f o b s e r v a t i o n s than k i s a t l e a s t MLE  does.  three; otherwise  Adaptive  Lemma 2 . 2 . 1 .  that i s greater  they g i v e t h e same e s t i m a t e  as the  e s t i m a t o r s w i l l a l s o be proposed i n S e c t i o n 3.  (Hudson [ 1 9 7 4 1  Suppose Y ^ P o i s s o n  ?  Peng  [1975]).  Cu) and G : R -* R i s a measureable f u n c t i o n such  that E | G ( Y ) I < oo and G ( y ) = 0 i f y < 0. Then y E G ( Y ' ) -= E Y G ( Y ^ 1 ) . v y u The f o l l o w i n g lemma .gives the unbiased e s t i m a t e A o f the d e t e r i o r a t i o n o i n r i s k o f X = X + f ( X ) as compared t o the r i s k o f the MLE X.  Lemma 2 . 2 . 2 .  (Hudson I1974J, Peng  [1975]).  Suppose X i s a random v e c t o r w i t h  independent P o i s s o n random v a r i a b l e s  as c o o r d i n a t e s , and X i s the c o r r e s p o n d i n g  P o i s s o n parameter v e c t o r .  Then  the d e t e r i o r a t i o n i n r i s k D o f X = X + f ( X ) as compared to t h e r i s k of the MLE  i s D = R(X,X) - R(X,X) = E, A where X o A  Proof:  o  =  J  P P I f .(X) + 2 ,S. X . [ f . ( X ) . , l i i i=l 1=1  Use Lemma 2 . 2 . 1 (see Peng  -f.(X-e.)]. l l  [1975]).  We see t h a t a s u f f i c i e n t c o n d i t i o n f o r an e s t i m a t o r  o f t h e form  X + f ( X ) t o have s m a l l e r r i s k than X (under squared e r r o r l o s s ) i s t h a t  A  P o P E f . ( x ) + 2.E x. [ f . ( x ) - f . ( x - e . ) ] 1=1 1=1  = °  1  1  1  +P <^ 0 f o r a l l x e J with s t r i c t  i n e q u a l i t y f o r some  _+ x e J' . P  The f o l l o w i n g  theorem i s due t o Peng  [1975].  Theorem 2.2.3. L e t X^,...,Xp be independent P o i s s o n random v a r i a b l e s w i t h unknown e x p e c t a t i o n s A^,...,A , and l e t t h e l o s s f u n c t i o n L be g i v e n by * L(X,X) =  P o ? E (A. - X.) . 1-1 ' 1  The  1  estimator X - [ ( p - N - 2 ) / S ]H Q  dominates t h e MLE, X i f p >^ 3.  +  Here  X =• (X_,.. ., X ) , 1 P X = (X.,...,X ) e [Q,°o]P, 1 p X. H. = ..E (1/k) f o r i = 1 k=l 1  1  p (H. = 0 i f X. = 0 ) , 1  X  H = (H, ,...,H ) , P 1  S =  l l H l .2  2 E H., i=l P  =  1  N  o  = # {X. : X. = 0} = t h e number o f zero o b s e r v a t i o n s , and (p-N - 2 ) , 1 x o + r  Max {p-N -2,0;}. o Proof:  Use Lemma 2.2.2 (see Peng ;[1975]).  14  In a d d i t i o n to the squared e r r o r l o s s f u n c t i o n , we s h a l l c o n s i d e r loss functions.  other  We d e r i v e an i d e n t i t y s i m i l a r t o t h a t o f Lemma 2.2.2.for  the case o f "k-normalized squared e r r o r l o s s " P  L (A,A) K  =  Z  (k-NSEL),  ^ 2 k ( A . - 2^) A *  i=l where k i s a p o s i t i v e i n t e g e r . discussed  i n S e c t i o n 4.3.  The m o t i v a t i o n  f o r c o n s i d e r i n g k-NSEL i s  The d e r i v a t i o n o f t h e i d e n t i t y w i l l be decom-  posed i n t o t h e f o l l o w i n g f o u r  lemmas.  Lemma 2.2.4. Suppose Y * P o i s s o n  (y) and h ; J -*• R i s a real^-valued f u n c t i o n such  that (1)  E^  |h(Y)J  (.2)  h(j) =0  < -  i f j < 0.  Then v  hOQ_ _ _ h(Y+l) u -u y Y+l  Proof:  v  v  y  =  0  ti.  y!  y=l  V  = o + _  I  E  y  s  h(y+i)\;f- y  y=0  h(y l) +  ' y=o =  -y  oo  ^  .  y  yJ  eV y !  h(Y+l) Y + l -' Q.E.D.  The  next lemma i s an immediate consequence o f Lemma 2.2.4.  15  Lemma 2.2.5. Suppose Y ^ P o i s s o n  ( u ) , k i s a p o s i t i v e i n t e g e r and h : J -> R i s  a r e a l - v a l u e d f u n c t i o n such  that  (1)  E  |h(Y+j)|  (2)  h ( j ) = 0 i f j < k.  y  < ~, j = 0 , . . . , k - l  Then  v y  Proof:  H(Y) ^k  h(Y+k) (Y+k)  =  %  =  ( k )  y  h(Y+k) (Y+k)•(Y+k-1)•••(Y+l)  I n d u c t i o n on k and a p p l i c a t i o n o f Lemma 2.2.4.  Lemma 2.2.6 below i s a g e n e r a l i z a t i o n o f Lemma 2.2.2 to t h e v e c t o r case. Lemma 2.2.6. independent Suppose  ^  positive integer.  If f  f o l d C a r t e s i a n product (1) (2)  Poisson :J  P  (A X, i = l , . . . , p , p >^ 2, and k i s a  R, i = l , . . . , p , a r e f u n c t i o n s on t h e p-  o f J,, such t h a t  E | f , ( X + je,);| ••:«», j ••= 0 , . . . , k - l f ( x ) = 0 i f x < k, ±  then f .(X) E, A  Proof:  f ; ( X + ke.) = E  A l  k  X  1  1  (X. + k ) l  C o n d i t i o n on  (  k  )  X^ : j ^ i  and a p p l y Lemma 2.2.5.  L e t X = X + f ( X ) be an e s t i m a t o r o f X, where f ( X ) = ( f , ( x ) , . . . , f (X)) 1 P and  t h e f . ' s s a t i s f y t h e c o n d i t i o n s i n Lemma 2.2.6. l  g i v e s an unbiased  The next  lemma  e s t i m a t e o f D^, t h e d e t e r i o r a t i o n i n r i s k o f X as  compared t o the r i s k o f X.  16  Lemma 2.2.7. Under k-NSEL, t h e d e t e r i o r a t i o n D  k  i n r i s k of A i s  = R(A,A) - R(A,X) =  A. fc  where  A k  =  p  f ( X + ke.)  P  E  —  E  2  77V + 2  1-1 1 X  (  (x +  k  1  1-1  )  k)  1  f . ( X + ke.) - f . ( X + (X. + k) — (X + k ) i  1  1  (  U  (k-l)e.)  1  +  k  k  1  )  ;  Proof: p R(A,A) = E £ A .  (A. - X. - f . ( X ) ) 1  1  k A.  1=1  P = E, E A. . -  :1=1  2  1  (A  - X ) 1  ^ - T ;  .k A.  2  + E,  —  A  p  f (X)  E  +  p  2  1=1  A.  k  2E,  (X -A.) f . ( X )  E  X . ..  1=1  P f (X) p X.f.(X) = R(A,X) + E, E - i = — + 2E. E A . .. k A . . . k i=l A. 1=1 A. i l 2  ,k A.  p f (X) 2E. E -=r-T A . .. ,k-l 1=1 A. ;  v  l  p f ( X + ke ) [ E -± p£r1-1 (X. + k ) 2  = R(A,X> + E  W  p + 2  E  (X  1=1  The l a s t e q u a l i t y f o l l o w s  f .(X + ke.) -. f . (X + ( k - l ) e . ) . + k) 1 Ijr=-], ••' " (X. + k) W  from Lemma 2.2.6.  The r e s u l t f o l l o w s  immediately.  Q.E.D. The s p e c i a l case o f Lemma 2.2.7.when k = 1 i s t h e case when n o r m a l i z e d squared e r r o r l o s s i s c o n s i d e r e d . below.  The r e s u l t i s s t a t e d  as a c o r o l l a r y  17  Corollary  2.2.8.  Under the n o r m a l i z e d squared e r r o r in  r i s k of A i s D  = R ( A , A ) - R(A,X) = E  + 2  The f o l l o w i n g  P Z 1=1  loss  the  deterioration  A , where  [f.(X+e.) - f.(X)']. 1  1  theorem of Clevenson and Z i d e k [1975].provides a c l a s s  of e s t i m a t o r s A of A dominating error  loss function,  the MLE  under the n o r m a l i z e d  squared  function.  Theorem 2.2.9. L e t X^,...,Xp be p independent parameters A ^ , . . . , A p  Z =  Z X.. 1=1 function  P  (p >_ 2) .  P o i s s o n random v a r i a b l e s w i t h unknown  Let A = ( A ^ , . . . , A  P  ), X =  (X^,...,X ) , and P  L e t the l o s s f u n c t i o n be :the n o r m a l i z e d squared  loss  1  L(A,A)  =  z 1=1  (A.1  A.) /A;.; 2  1  1  Then, f o r a l l A , the r i s k u s i n g the e s t i m a t o r  A* = [ 1 -  (<j>(Z)/(Z+p-l)) ]X  is  l e s s than or e q u a l to the r i s k u s i n g A  is  n o n d e c r e a s i n g and n o t i d e n t i c a l l y z e r o .  = X when <f> : [0,°°) ->  Proof:  L e t f . ( x ) = -cb(z)x./(z+p-l)  x  X  =0  i f x. > 0  x — ' i f x. < 0, i = 1  x  ,...  >P-  [0,2(p-l)]  By C o r o l l a r y 2.2.8, the d e t e r i o r a t i o n i n r i s k i s  A  =  1  -  1  p E  <p (z+l)(x.+l) 2  1  1=1  t _ i _ _ \  (z+p)  2 < — — ( ^ ^ — — z+p +  = <p(.z+l)  2 2  p E  (f)(z+l)(x.+l)  1=1 •  + 2  -~~ 1  Z  P  +  2(j)(z+l) + 2 4l(z+Jil^. ( i z+p—± s  n  c  e  p E  § ±  s  1  1=1  •  <f>(z)x. , -  i  z+p-1  nondecreasing)'  ,<Kz+D _ 2(p^-l) z+p  z+p-1  <_ 0 ( s i n c e 0 <_ <f>(z+l) £ 2(p-l););. A  T h e r e f o r e , R(A,X) >_R(A,A) f o r a l l A.  A  I n o t h e r words, A dominates X  u n i f o r m l y i n A. In S e c t i o n  Q.E.D. 4, we propose e s t i m a t o r s A o f the form X + f ( X ) which  s a t i s f y the i n e q u a l i t y A^ £ 0 f o r a l l x and A^ jjf 0. therefore  dominate the MLE under k-NSEL.  Those e s t i m a t o r s  19  SECTION 3.  3.1  ESTIMATION UNDER SQUARED ERROR LOSS  Introduction P r o b a b l y the most e x t e n s i v e l y s t u d i e d l o s s f u n c t i o n used i n estima-r  t i o n problems i s the squared e r r o r l o s s f u n c t i o n .  I t s p o p u l a r i t y can be  a s c r i b e d to i t s mathematical t r a c t a b i l i t y and a l s o to the f a c t t h a t i s an a c c e p t a b l e a p p r o x i m a t i o n i n a wide v a r i e t y of s i t u a t i o n s "  "it  (DeGroot  [1970], p. 228) where the l o s s depends s o l e l y on the d i f f e r e n c e between a parameter and i t s e s t i m a t e .  I f we a l l o w k to be e q u a l to z e r o , the  squared e r r o r l o s s f u n c t i o n can be thought of as a s p e c i a l case of k-NSEL. T h i s s e c t i o n w i l l be l i m i t e d i n scope to simultaneous e s t i m a t i o n of P o i s s o n parameters under  the squared e r r o r l o s s f u n c t i o n . The independent  i s as d e s c r i b e d p r e v i o u s l y . i  = l,...;p  As mentioned  and  that  We one  suppose t h a t  <v  P o i s s o n (A^),  o b s e r v a t i o n i s taken from each p o p u l a t i o n .  i n S e c t i o n s 1 and 2, t h i s problem has been i n v e s t i g a t e d by  Peng [1975], who  succeeded i n f i n d i n g e s t i m a t o r s dominating the MLE when  p >_ 3 (Theorem 2.2.3).  B a s i c a l l y , h i s proposed e s t i m a t o r s p u l l the  towards the o r i g i n whenever the number of non-zero o b s e r v a t i o n s two.  setting  MLE  exceeds  The performance o f h i s e s t i m a t o r s i s expected to be good when the  u n d e r l y i n g parameters A^ a r e r e l a t i v e l y s m a l l . meters a r e l a r g e , however, v e r y l i t t l e anticipated.  improvement over the MLE i s  I n t h i s s i t u a t i o n , some v e r y l a r g e o b s e r v a t i o n s a r e l i k e l y  to o c c u r and b o t h Peng's e s t i m a t o r virtually  When some of the p a r a - :  the same e s t i m a t e .  [1975] uses S t e i n ' s method  (Theorem  2.2.3). and the MLE  give  I n o r d e r to remedy t h i s s i t u a t i o n , Peng  [1974] t o modify h i s e s t i m a t o r .  I f a l l the parameters A_ a r e . r e l a t i v e l y l a r g e , none of the e s t i m a t o r s proposed by Peng w i l l g i v e n o t i c e a b l e improvement over the MLE.  Basically,  20  t h i s i s due t o t h e f a c t t h a t those  estimators  o r i g i n , a p o i n t f a r away from t h e t r u e X. observations  are biased  Estimators  toward the  that s h i f t the  towards a p o i n t i n a neighbourhood o f t h e t r u e u n d e r l y i n g  parameter would be expected t o g i v e b e t t e r e s t i m a t e s  i n t h i s case.  For  each non-negative i n t e g e r k, we show t h a t t h e r e i s a f a m i l y o f e s t i m a t o r s " (k) (k) X of X such t h a t X A  dominates t h e MLE u n i f o r m l y A  error loss function. property  F o r a f i x e d k, t h e e s t i m a t o r  i n X under t h e squared (k)  X  has the  t h a t i t p u l l s the MLE towards t h e i n t e g e r k whenever t h e number  of o b s e r v a t i o n s  x^ g r e a t e r than k i s a t l e a s t t h r e e .  the same e s t i m a t e  as t h e MLE.  Otherwise, i t g i v e s  I n t h e case when k=0, we have Peng's  result. Estimators  that s h i f t  the MLE towards a p o i n t determined by t h e  d a t a i t s e l f w i l l a l s o be proposed.  These a d a p t i v e  estimators are  expected t o perform w e l l f o r a wide range o f parameter v a l u e s , i n c l u d i n g the case when t h e parameters a r e a l l r e l a t i v e l y l a r g e b u t s i m i l a r i n value.  Most o f the r e s u l t s of Peng [1975] w i l l be g e n e r a l i z e d .  In the  next s u b s e c t i o n , we w i l l i n t r o d u c e t h e n o t a t i o n t h a t i s used i n t h i s section.  E s s e n t i a l l y , we employ t h e n o t a t i o n used by Peng  3.2 N o t a t i o n L e t x = ( x . . . , x ) be a v e c t o r o f o b s e r v a t i o n s 1 P l 9  vector  (X,,... ,X ), where the X.'s a r e m u t u a l l y 1 p l  random v a r i a b l e s w i t h parameters  [1975].  o f t h e random  independent  Poisson  respectively. Let  X^,...',X7,  f = ( f ^ , . . . , f p ) be as d e f i n e d i n S e c t i o n 2.1 and s a t i s f y t h e f o l l o w i n g conditions: (1)  f ^ ( x ) = 0 i f x has a n e g a t i v e  coordinate  (2) E. |f.(X+je.);| < co f o r j = 0,1-v A  The  1  1  n o t a t i o n employed here i s l i s t e d below f o r r e f e r e n c e .  21  Definitions: (1)  EL = #{x^  : x^ = j } , i . e . the number o f x^'s t h a t a r e e q u a l  to j .  (2)  P P. I = Max{x.}.; m = Min{x.} . i=l i=l 1  1  (3)  N = (N ,...,N ) . O X,  (4)  F o r any non-negative i n t e g e r k, r  (5)  h : J •+ R i s a r e a l - v a l u e d f u n c t i o n such t h a t h ( y ) = 0 i f y < 0.  (6)  S =  = (p -  k E N n=0  - 2) .  P 2 E h ( X . ) ; S = E. ti ( x . ) . i=l i=l p  2  1  1  (7)  W r i t e f ( x ) = T (N) i f x. = j .  (8)  <J>: J  ±  P  •> R i s a r e a l - v a l u e d  properties:  ( i ) <j>  f u n c t i o n s a t i s f y i n g the f o l l o w i n g  i s n o n d e c r e a s i n g i n each argument x^  whenever x. > k.  l (ii) $  whenever X.  l (iii)  —  i s n o n i n c r e a s i n g i n each argument x^ < k.  —  There i s a r e a l number B > 0 such t h a t 0 <j <j>(x) 1 2B and <f>(x) i  0.  We a r e i n t e r e s t e d i n f u n c t i o n s h which s a t i s f y the p r o p e r t i e s listed  i n Lemma 3.2.2  representative  below.  B e f o r e s t a t i n g the lemma, we p r o v i d e a  example of the f u n c t i o n s h we  want.  r  22  Example 3.2.1. (a)  T o r k >_ 2 y-k Z n=2  h(y) = 1 +  i f y = k+2,k+3,  =M  i f y = k+1  = 0  i f y = k or y < 0  = -b  k-y ZZ r — - r — k+l-n n=l  if y =  0,...,k-l  where b i s a p o s i t i v e number t o be determined such t h a t  (8) o f Lemma  1 -1 One such b i s b = v3( X — - z ) k+l-n n=l k  3.2.2 below h o l d s .  (b)  For k = 1  h(y) = 1 +  Z T T ~ k+n n=2  i f y = k+2,k+3,...  0  = 1  i f y = k+1  =0  i f y = k or y < 0  = -b  if y = 0  where b i s any p o s i t i v e number, (c);  For k = 0 1 Z i n=l y  h(y) =  = 0 The f o l l o w i n g lemma g i v e s denote h. = h ( j ) . 3  i f y = 1,2,... i f y <_ 0. the p r o p e r t i e s o f h.  F o r s i m p l i c i t y , we  23  Lemma 3.2.2. L e t h be as d e f i n e d i n Example 3.2.1.  Then h s a t i s f i e s t h e f o l l o w i n g  properties: (1)  2 2 h. - h . , J j-1  (2)  j [ h ^ - h --^]  i s nonincreasing  i s nondecreasing  lim j [ h . - h  > h._  h  (4)  h  j = 1,2,...  (5)  h. > 0 i f j > k+1.  = 0.  k  3  -  (6)  I f k > 0, then h  (7)  h  (8)  3B  (9)  \  (10)  h  k  +  2  j  i n j f o r j > k and  ] = B f o r some B > 0.  (3)  v  i n i f o r i > k+1.  +  l  i  B .  > hjh  2  1  p r o v i d e d k > 0.  > j[h  - h  < 0 f o r j < k.  - h  j_] f o r j > k+2.  . < h - h f o r 1 < j < k i f k > 0. j - 1 - j+1 j 2  2  2  J  The p r o o f o f the lemma i s s t r a i g h t f o r w a r d and i s o m i t t e d .  3.3  S h i f t i n g the MLE Towards k We use the n o t a t i o n d e f i n e d i n 3.2 and d e f i n e  f.(x)  r cb(x)h(x ) ^ i-,  i = l,...,p.  (3.3.1)  * (k) We s h a l l show t h a t t h e e s t i m a t o r  X  = X + f ( X ) o f X dominates X  u n i f o r m l y i n X under t h e squared  e r r o r l o s s f u n c t i o n when p >^ 3.  The  24  " (k) estimator X provided  s h i f t s t h e c o o r d i n a t e s of t h e MLE towards t h e i n t e g e r k  t h e number o f o b s e r v a t i o n s g r e a t e r than k i s a t l e a s t  three.  R e c a l l t h a t t h e r i s k d e t e r i o r a t i o n o f t h e e s t i m a t o r X = X +f(X) as compared  t o X i s , by Lemma 2.2.2, R(X,X) - R(X,X) = E  A  A  P where A = E f . ( X ) + 2 E X . [ f . ( X ) - f . ( X - e . ) ] . i=l i=l In terms o f N and (N) , A can be r e w r i t t e n as P  2  1  A =  1  E N.¥.(N)+2 E j_Q J 1 j  1  =  jN..(N) l ^  where 6^ i s an (£+1)-vector w i t h t h e j other c o o r d i n a t e s z e r o .  1  th  (3.3.2)  1  - ¥._ (N-6.+6._ ) ]  (3.3.3)  c o o r d i n a t e e q u a l t o one and t h e  * (k) To show t h a t X = X + f ( X ) dominates X under  the squared e r r o r l o s s , i t s u f f i c e s t o show t h a t A(x') <_ 0 f o r a l l +P x e J  .  The p r o o f w i l l u s e t h e f o l l o w i n g s e r i e s o f lemmas. F o r  convenience, we d e f i n e A j  .  J  H  [  T  .,<,»_««,]  W  ( 3  ^  4 )  (= x . [ f . ( x ) - f . ( x - e . ) ] , w i t h x. = j ) i i I I I 7  Lemma 3.3.5. (1)  F o r k >_ 2, j [ h . - h. ] $ ( x ) r , N . A. < - — J 3=± ^ S 3  (2)  A. <  <p(x)kN r h , fc  S + h  Ll  1  .  S - h.(h.+h. ,) [ \ .1 j " ] i f 1 < j < k. S - h f +:-hT... J J-1 1  25  (3)  j [ h -h - -  A. <  1  ] r cJ,(x)N  11  K  S - h (h +h ) 1 -1 - j ] S - h. + h f ,  [  1  1  S  3  A[h -h„ J U ( x ) N . A  £ 1  k  [  1  S-2h  —  2  V " ] for k+2 < j <  Proof: (1)  i s t r u e because <j>(x) i s n o n i n c r e a s i n g f o r x^ <_ k, i = l , . . . , p .  (2)  i s t r u e because cb(x) i s n o n i n c r e a s i n g f o r x. <^ k and h,  (3)  n  < 0.  "..is t r u e f o r the f o l l o w i n g r e a s o n s : (i)  ? I f r. > 0, then S > 2h. f o r k+2 < j < I and N. f 0. k J ~ 3  2  2  (ii) (iii) (iv)  - hj_i^  n o n i n c r e a s i n g by (1) o f Lemma 3.2.2.  s  lu > n cb(x) i s n o n d e c r e a s i n g  when x^ >^ k, i = l , . . . , p . Q.E.D.  Lemma 3.3.6 2h  2  > h  k+1 ~  2  x,  - h  2  £_ 1  where H > k+2.  —  Proof: By (9) o f Lemma 3.3.2, h ^ ^ ^_ j [h +  l u - h j _ ^ i s decreasing,  < -  Hence  2 2h.  < 3  . k+1  2  V  —  [ h  > j [ h . - h. J — J J-1  i J  +  l  Vl  ]  , •' J  - j_i^ n  f ° J — k+2. r  Also, since  26  for j>  h. , 3-1  k+2. Q.E.D.  3.3.7.  Lemma  2h 2 V " ^ k' " k S - h£ + h^_  £  k+l  {N  +  E  k + i  2  S -  i j=k+2 N  }  3  r  r  >  k  k  1  Proof:  The  l e f t hand s i d e o f t h e above i n e q u a l i t y i s A  N  \+l  +  +  2h  2  S -  I.  ._ .  W  j=k+2  N  ^  j _  . 2 . .2  S - h +h^_  J  £  1  £ W  m  -  W r tl> h  h  S  +  "£  +  h  £ - 1=k+2 ^ H 2  jJ+2 V h  t  l l .  £  £  . i=k i V  W ^ r h f o l i > -2 4 + i 2  +  +  s  k^  -' h  r  >  x S  N  ~ £ h  + h  5—  (since  + h  N  i ? h  L i  2 2h,  2  _ £L-h  0  2  - h  0  , by Lemma  3.3.6)  £-l  >_ r ^ ( s i n c e r ^ > 0) Q.E.D.  Theorem  3.3.8. r (|)Cx)hCx ) ±  With f . (x) = i n Lemma 3.2.2, A < 0. .  1  , i = l , . . . , p , p s 3, and h as d e s c r i b e d  Proof:  E N.Y.(N) + 2 E jN.[¥.(N) - V. .(N-g.+fi. , ) ] j=0 j=l J J J 3 J  R e c a l l that A =  3  Case 1.  r, = 0. k r, > 0. k  Case 2.  (i)  E j=0  (ii)  2  -  1  _  1  Then A = 0 < 0. —  2 2 r, cb (x)  _ NX ( N )  I  3  3  3  s  £ k-1 E A. = 2 E A. + 2A^ + 2A^ j-1 3=1 3  + 2  3  w i t h the u n d e r s t a n d i n g  £ E A. J=k+2 3  k-1 E A. = 0 i f k <_ 1.  that  j=l  J  By Lemma 3.3.5,  2  k-1 E A  3=1  <t>(x)r -  <_  k-1 E j [ h - h ]N  3=1  3  4>(x)r -  1  3  3  -  1  S - h.(h.+h [ ] S - h + h ^ 3  3  2  k-1  *  2  S - h j  j=l  [  h  j J  j - i  h J  s i n c e h. < 0 f o r j < k and h. > h. J 3 J-1  ]  N  j  [  ) ]  3  2  - h h  A, A S — h . ~f* h, i J 3-1 1  g  so L h > h.h. . f o r 1 < j < k 1 0 - ] J-1 n  J  2 2 > 0 i m p l i e s t h a t S >_ 1 3B . By (8) o f Lemma 3.2.2 2 k"l we have S - h. i . - h..h > 0 f o r N. ? 0. Hence E A. < 0. A g a i n by 3 1 0 3 3=1 ~ Now r  3  f c  n  k  +  1  n  3  Lemma 3.3.5 ^(x)kN r h _ k  \  -  q  . 2  S + h  (because h  k  k  h  k-l  = 0).  k  (k+DN  1  a  n  d  V l  k + 1  S  r^(x)h  28  As a r e s u l t ,  2kN r <Kx)h  £ 2  Z  j  =  1  A. £  k  J  k  "  k  1  V l  S +  2r <(>(x)  "  [  (  ±  k  1  £ )  k  N  l \  +  l  +  * V £-l  +  [  h  3  (by Lemma 3.3.5. (3)' ) . £  2kN  k V  k  2Kx)r kN  1  k  2  k + 1  h  =k+2  j— 2 - 2 - 1 S-h +h _ £  h  k  +  A  +  l  +  £ [ h  £" £-l h  S-2h  2kN r Kx)h _ k  k  <; + h  S  2^(x)r kN  1  k  2  * j „ 2 2 j=k+2 S-h +h _  ]  N  1  h  + h  ji  1  1  k + 1  £ [  k  £  S  2r <p(x) "  N  Thus,  (x)h _  i —  S - 2h?  *  ]  k + 1  h  £  11  k + 1  S  2  \ - l  +  2r <l>(x)£[h - h . J S  £ S-2h [N i + Z N. ^ ] "k+1 .... 2.. 2. =k+2 S-t^+h^ 2  k L  3  (since h  2kN rk»(x)h k  1  -  S  l  k + ]  _ > Alh^-h^])  2»(x)r kN  k M l  k  k + 1  h  _  k + 1  2r <Kx)B 2  l  (by Lemma 3.3.7 and s i n c e £[h -h _^] ^ B) . Consequently, 2kNr^ A £  ( x )  ~  s +K  h  Li  2kN -  k + I  r Kx)h ^ k  k + 1  r j ^ x ) — z [2B - (|>(x)J (3.3.9)  < 0 s i n c e 0 £ <J>(x) £ 2B, h ^ < 0, and h  k  +  1  Q.E.D.  > 0.  29  Our  r e s u l t s show t h a t to e v e r y f i x e d n o n - n e g a t i v e i n t e g e r k, t h e r e  corresponds  i (k)  a c l a s s of e s t i m a t o r s X  o f X which has t h e p r o p e r t y  that  all  i t s members p u l l the c o o r d i n a t e s o f the MLE towards k when t h e number % of o b s e r v a t i o n s t h a t a r e g r e a t e r than k ( i . e . E N.) i s a t l e a s t t h r e e . j=k+l 2  I f t h e number o f o b s e r v a t i o n s g r e a t e r than k i s l e s s than t h r e e , then the " (k) estimators X g i v e the same e s t i m a t e as t h e MLE X. The c h o i c e o f k is c r i t i c a l . I t may depend on p r i o r i n f o r m a t i o n a v a i l a b l e about the range A  of the X.'s. When k=0, the e s t i m a t o r s X l observations  (0)  s h r i n k a l l the non-zero  towards zero as l o n g as the number o f non-zero  i s a t l e a s t three.  observations  Such a v a l u e o f k should be chosen o n l y when we have  some p r i o r knowledge about the X^.'s i n d i c a t i n g t h a t they aire a l l c l o s e "(0) to z e r o . X i s expected  t o perform w e l l i n t h i s s i t u a t i o n ,  when a l l t h e X.'s a r e c l o s e t o one another. x suggests  I f the p r i o r  especially  information  t h a t the A_/s a r e l i k e l y t o be l a r g e and w i t h i n the range [a,b]  w i t h 0 < a < b, then a l a r g e v a l u e o f k somewhere around a may be chosen. Choosing k i n t h i s manner w i l l l i k e l y l e a d t o c o n s i d e r a b l e improvement over  the MLE.  Hence, having  some p r i o r knowledge about the parameters  X^ i s advantageous t o the e s t i m a t i o n problem.  I n f a c t , we see t h a t ,  a c c o r d i n g t o our s i m u l a t i o n r e s u l t s r e p o r t e d i n S e c t i o n 7, our Bayesian a n a l y s i s i n S e c t i o n 5 does l e a d t o s u b s t a n t i a l improvement over  the u s u a l  procedure when t h e parameters a r e known to be c l o s e t o one another. N o t i c e t h a t t h e bound  (3.3.9) f o r the unbiased  e s t i m a t e A o f the  " (k) d e t e r i o r a t i o n i n r i s k of X  depends on k, N^ and  s a v i n g s i n r i s k would be g r e a t e r i f N^ and  ...Hence,  are large.  In other  words, when more o b s e r v a t i o n s a r e c l o s e t o the chosen i n t e g e r k t h e r i s k w i l l be reduced  much more.  One might say t h a t r e l i a b l e p r i o r  information  30  can be p r o f i t a b l y e x p l o i t e d i n our e s t i m a t i o n problem.  The dependency o f  the bound f o r A on k, N. and N, ,- f u r t h e r i m p l i e s t h a t t h e e s t i m a t o r s k k+1 " (k) + X  f o r various k e J  are competitive;  one cannot dominate t h e o t h e r .  In t h e case o f simultaneous e s t i m a t i o n existence  o f an e s t i m a t o r  which s h r i n k s  t h e MLE towards zero and  dominates t h e MLE i m p l i e s t h e e x i s t e n c e towards any f i x e d p o i n t and s t i l l  o f p normal means, t h e  o f another e s t i m a t o r  dominating t h e MLE.  shrinking  T h i s i s due t o  the t r a n s l a t i o n i n v a r i a n c e o f t h e normal d e n s i t y and t h e squared e r r o r loss function. property  However, i n our c a s e , we do n o t have t h i s  f o r the Poisson  of b e t t e r e s t i m a t o r s  invariance  p r o b a b i l i t y f u n c t i o n , and hence t h e e x i s t e n c e  s h r i n k i n g towards a p o i n t other  automatic even though a b e t t e r e s t i m a t o r  than zero  i s not  s h r i n k i n g towards zero e x i s t s .  Thus our r e s u l t s a r e n o t obvious consequences o f Peng's. Since our e s t i m a t o r s  depend on h, i t i s i n t e r e s t i n g t o f i n d more  examples o f f u n c t i o n s h which have t h e p r o p e r t i e s i n Lemma 3.2.2.  Below  are some examples. Example 3.3.10. (k = 0) L e t h(y) = In(ay)  i fy > 1  = 0 where a •> 4.  i fy < 1  L e t B = 1.  the p r o p e r t i e s l i s t e d (a)  I n order  We check below t h a t t h i s f u n c t i o n h s a t i s f i e s  i n Lemma 3.2.2.  t o show t h a t p r o p e r t y  2 1 2 to show t h a t h.. >_ ^ ^ j j - i h  +  h  2 j+l^ 2  the f u n c t i o n G(y) = [ I n ( a y ) ] (b) = y[ln y  Property ln(y-l)]  (2):  f  o  r  J  - * 2  (1) h o l d s , T  h  i  s  i  s  c  e  i t i s sufficient r  t  a  i  n  l  y  true  since  i s concave f o r y >_ 1.  The f u n c t i o n F(y) = y [ l n ( a y ) i s decreasing  I n a(yr-l)]  when y>_ 2 s i n c e the d e r i v a t i v e  31  F' (y) = l n ( l + J  >^ 2 [ l n ( 2 a )  - - \ < 0 f o r y > 2. y-1 —  y-1  - I n a] whenever a >_ 4.  A l s o , h. - h = I n a 1 0  Moreover, l i m j [ l n j - l n ( j - l ) ] = 1  = B. (c)  Properties  properties this  ( 3 ) , (4), (5), (7), and (9) c l e a r l y h o l d , and  (6), ( 8 ) , and (10) a r e not a p p l i c a b l e here because k = 0 i n  example.  Example 3.3.11. (k = 0) L e t h : J -*- R be any f u n c t i o n such t h a t h^ = 0 i f j <^ 0 and 1 h. = E — f o r i = 1,2,... . J i g n = l °n satisfying j  J  (1)  g  (2)  g  (3)  {—}  Here {g :}. i s a sequence o f r e a l numbers n  = 1  x  n  +  1  - g  n  > 1, f o r n = 1,2,...  i s nonincreasing  and l i m ^-  g  Properties (a)  .  (1) through Property  = B > 0.  g.  (10) o f Lemma 3.2.2 a r e checked as f o l l o w s :  2 2 (1) h o l d s s i n c e 2h_. - (hj_-j_  +  j-1 i SjSj+l  { 2 [ g ^ i - g j nt = 2 l ^r j  +  1  J  n  n  2 j+i^ g  J  i  1 - + 8j „ 2  +  8,  1  -g— - > > J  j  +  0  1  f o r j >_ 2. (b)  Property  (c)  A l l t h e other  Property  (2) h o l d s s i n c e  s a t i s f i e s requirement  properties clearly  (8) o f h g i v e n  (3) above.  hold.  i n Lemma 3.2.2 guarantees t h a t  S - h.(h. + h._^) >^ 0 f o r j < k, which i s a s u f f i c i e n t c o n d i t i o n  that  k-1 £  A. <_ 0.  However, i t i s n o t a n e c e s s a r y c o n d i t i o n , as t h e f o l l o w i n g  j=l theorem shows. 3  32  Theorem 373'.'121 L e t h : J -> R be as d e s c r i b e d (3) and (8) a r e r e p l a c e d  i n Lemma 3.2.2  except t h a t  properties  by  ( 3 ) ' h. > h . , f o r j > k+1 3 J-1 (8)' h j = -b < 0 f o r j = 0 , . . . , k - l . Then Theorem 3.3.8 Proof:  still  The change s t i l l  holds. gives  k-1 E A. <_ 0. j=l 2  The f o l l o w i n g theorem i s a s l i g h t v a r i a t i o n o f Theorem 3.3.12.  Theorem 3.3.13. L e t h be as d e s c r i b e d 0 <_ cb(x) <_ Min{2B,l} i f x r f .(x) =  i n Theorem 3.3.12. < k, i = 1,.. . ,p.  ±  cb(x)h(x ) —  l  Suppose Define  i f x. > k  S  l  =0  i f x. = k  i k = cb(x) Min{—-—, 1} i f x. < k b r  *(k) + A = X + f ( X ) s a t i s f i e s A(x) £ 0 f o r a l l x e J  Then the e s t i m a t o r (i.e.  A  for i = l,...p.  dominates X u n i f o r m l y  P  under the squared e r r o r l o s s function).  Proof: It  can be checked t h a t  k-1 E A. < 0 f o r k > 0 and t h a t J-1  "  rV(x)  I  E  j-0  J  < 3  3  ~  ;  .  Hence A < 0 as shown i n Theorem 3.3.8.  s  Q.E.D. Note t h a t e s t i m a t o r s have the p r o p e r t y  A  of A g i v e n  i n Theorems 3.3.8  and 3.3.12  t h a t they p u l l the x^'s t h a t a r e f a r t h e r away from k  33  more than those x.'s  t h a t a r e c l o s e r to k.  1  observations experience  a great deal of s h i f t i n g , while the observations  close to k are s h i f t e d There i s s t i l l  T h i s means t h a t t h e extreme  to a l e s s e r  another  extent.  c h o i c e o f h t h a t w i l l guarantee A <_ 0.  Theorem 3.3.14. Let  h be as d e s c r i b e d i n Lemma 3.2.2 except  that property  (8) i s v .  r e p l a c e d by  (8)"  3h  2 + 1  >  h l  h . Q  Then Theorem 3.3.8 s t i l l h o l d s .  ( I n t h i s case,  |hg| has a l a r g e r upper  bound). Proof:  k-1 E  Note t h a t  j=l  A. i s s t i l l  l e s s than zero i n t h i s  case.  3  An example o f a f u n c t i o n h d e s c r i b e d i n Theorem 3.3.14 i s g i v e n below. Example 3.3.15. Let  (k >. 2)  J 1 h. = E n=l  i f j > k+1  = 0  if j = k  . k+l-n n=l L  A  The l^  1  estimators A  if  0 < j < k. -.  (k) d e r i v e d thus f a r have t h e p r o p e r t y t h a t i f t h e  o b s e r v a t i o n i s e q u a l to k, then A ^ ) =  s h i f t i n g o f the o b s e r v a t i o n s h a v i n g  That i s , t h e r e i s no  the v a l u e k.  The next  theorem p r o v i d e s  an e s t i m a t o r o f A which improves on t h e MLE b u t whose e s t i m a t e o f A^ th is The  not n e c e s s a r i l y equal to k i f the i  o b s e r v a t i o n i s e q u a l t o k.  theorem u n i f i e s and g e r n a l i z e s Theorems 3.1 and 5.1 o f Peng 11975].  34  Theorem 3.3.16. L e t h be as d e s c r i b e d and  In Theorem 3.3.12 except t h a t p r o p e r t i e s  (7) of h a r e r e p l a c e d (4) ' h  k  (7).' h  k  (4)  by  = -b +  > Max{l,B}.  1  Define r |) ( x ) h ( x ) f . (x) = - • — i^"" S k<  ±  :V:  :  = (j>(x) Min{ —  1  "  +  i f x. > k  x  1  }  ifx  < k  for i = l,...,p.  Suppose 0 <_ <p(x) <_Min{l,2B} i f x^ < k, i = l , . . . , p , and l e t  X  = X + f GO .  Then A G O < 0 f o r a l l x  The p r o o f of the theorem i s s i m i l a r t h a t we denote V (N) = f ^ G O  J  e  .  to t h a t of Theorem 3.3.8.  Recall  i f x^ = j .  Proof: k k+l F i r s t , note that 1 - — > 0 since r  S > (p -  Z N )h n=0  Case 1.  r, < 0. k —  Case 2.  r, > 0. k  n  2 + 1  >  r h k  h  k + 1  Then A = 0  ( 7 ) ' h o l d s and  .  < 0. —  2 2 £ „ r cp G O (i) Z N.W,(N) < j=0 3  (ii)  Z A j=l  3  S  < <p(x)  3  Z jN.[V(N) - V(N-6.+6. -)•] < 0 j=l ~ 3  k where V(N) = Min { — o 1 < j < k. b r  r  , 1 -  3  k\+l *} x  13  .  3  L  Note t h a t V(N) = V(N-6.+6. ,) f o r j J-l  35  (iii) 2 j  A  j=k+l  - 2(k l)N +  (Ji) . V ^ k + l  I?  k + 1  + 6 )J + 2 j k  J=k+2  J  1  2 H  viW  N )  -  2 ( k + i ) N  k i\  ( N  +  - k i 6  (the  r e a s o n i n g i s s i m i l a r t o t h a t o f Theorem 3.3.8).  (i),  ( i i ) , and ( i i i ) imply t h a t 2 k  \+l k r  < K x ) h  + 6  +  V*  k+l  A J  k )  0 0  (3.3.17)  Remarks: (1)  The bound f o r the unbiased  g i v e n by (3.3.17) depends on k and N  e s t i m a t e o f the r i s k d e t e r i o r a t i o n A k +  ^  .  If N  k +  ^ i s l i k e l y t o be l a r g e ,  " (k) i fA  1  then g r e a t improvement i n r i s k w i l l r e s u l t  i s used i n s t e a d o f X.  Moreover, t h e dependence o f (3.3.17) on k i m p l i e s t h a t the e s t i m a t o r s " (k)  +  1  A  f o r d i f f e r e n t values of k e J (2)  are competitive.  The s p e c i a l case when k = 0, cb(x) = 1, b = 1, and h i s as g i v e n  i n Example 3.2.1 (c) i s Peng's  [1975] Theorem 5.1, which s h r i n k s a l l  non-zero o b s e r v a t i o n s towards zero w h i l e a p o s s i b l y non-zero e s t i m a t e o f A. i s g i v e n f o r x. ="0. i i (3)  The case when k = 0, <j>(x) = 1, b = 0, and h i s as g i v e n i n  Example 3.2.1 (c) i s Theorem 3.1 o f Peng  [1975], which we s t a t e d as  Theorem 2.2.3 i n S e c t i o n 2. (4) x^ = 0.  I f b = 0 i n Theorem 3.3.16, A^ w i l l be e s t i m a t e d as zero if';-., However, i f b > 0, the e s t i m a t e f o r A^ w i l l be p o s s i b l y non-  zero i f x^ = 0.  The c h o i c e of r e l a t i v e l y l a r g e v a l u e o f b can be  i n t e r p r e t e d as r e f l e c t i n g t h e b e l i e f  t h a t the X^'s a r e non-zero.  36  3.4  Adaptive  Estimators  The e s t i m a t o r s A " ~ o f A suggested prechosen  non^negative  i n 3.3 p u l l t h e MLE towards a  i n t e g e r k, and t h e c h o i c e o f k i s guided by t h e  p r i o r knowledge o f t h e A.^'s.  A n a t u r a l q u e s t i o n which a r i s e s i s : I s  there.an. .estimator A o f A which s h i f t s t h e o b s e r v a t i o n s towards a p o i n t determined  by the d a t a i t s e l f ?  We s h a l l show t h a t t h e answer i s a f f i r m -  tive. P  R e c a l l t h a t m = M i n {x.}. i=l = l , . . . , p as f o l l o w s :  P We d e f i n e new f u n c t i o n s H. : J  1  i  x. -m E - f . „ m+n n=2  R,  1  1  H.. (x) = 1 + x •  i f x. > m+1 and m > 0 x —  =1  i f x. = m+1 and m > 0 x —  =0  i f x. = m and m > 0 x —  =0  ifm < 0  The f u n c t i o n s 3.2.2.  1  (3.4.1)  f o r i = 1,  have s i m i l a r p r o p e r t i e s as those of h d e s c r i b e d i n Lemma  We s t a t e t h e p r o p e r t i e s of t h e H^'s i n t h e f o l l o w i n g lemma.  Lemma 3.4.2. L e t B = 1.  The H 's,.i = l , . . . , p , have t h e f o l l o w i n g p r o p e r t i e s :  2  2  (1) BL(x) - H^(x - e^) i s n o n i n c r e a s i n g i n x ^ f o r x^ > m+1 w i t h m >_ 0. (2) x.[H.(x) - H.(x i- ..e.)J::is n o n i n c r e a s i n g i n x. f o r x. > m w i t h . x x x x x x m > 0 and l i m x.[H.(x) - H.(x - e . ) ] = B. — x.-*° i i i i x (3) H ( x ) > H ( x - e_^) i f x^ > m and m >^ 0. i  i  (4) H.(x) = 0 x  i f x. = m. x  (5) H.(x) > B x —  i f m > 0 and x. = m + 1. — x  37  (6) H.(x)  1  x. = m +11, i Proof:  > (j + x ) [ H . ( x + j e . ) « H.(x  and m > 0.  11  1 1  +  ( j - l ) e . ) ] f o r j > 1, —  2_  The p r o o f i s s t r a i g h t f o r w a r d and hence i s o m i t t e d . T i n 1  The f o l l o w i n g theorem p r o v i d e s us w i t h a c l a s s of e s t i m a t o r s X of X which p u l l the MLE  towards a p o i n t determined  by the d a t a , namely  the minimum of the x . ' s . l Theorem 3.4.3. Let X -f l , LmJ = X. A. I l m  v  [ m ]  = d | , . . . , A ^ ) be such t h a t m ]  m ]  (p-N -2) <p(X)H (X) m + I . , I = l ,». .i . ,»f» p i, where p E Hf (X) i=l The H..'s a r e as d e s c r i b e d i n Lemma 3.4.2. l 1  (1) (2)  N  (3)  <p(x) i s non-negative  (4)  <p(x) i s nondecreasing  m  = #{i  : X. = m} l  and p > 4. — and  cp(x) <_ 2B f o r some B > 0.  i n each agrument  x^.  ~ Tml  Then f o r a l l X = (X^,.-. .,X  ), X  dominates X under squared  error loss.  Proof: 2 D e f i n e f.(N) = f . ( x ) i f x. = j > 0 and l e t S = E HT(x). 3 i-1 p r o o f i s v e r y s i m i l a r to t h a t of Theorem 3.3.8. Case 1. (p-N - 2 ) , = 0. Then A = 0 < 0. m + — Case 2. (p-N - 2 ) , > 0. m + P  1  (i)  I E j=m  ?  N.¥,CN) ^ 2  2  (p-N  1  -2)J$ (x) 2L=-J£ 2  b  The  38  (ii)  £ 2 Z A. = 3=m 3  £ 2 Z A. j=m+l 3  . 2 (m+l)N = -  m  +  m 1  Cp^N -2) cb (x)h(m+l) + 2 m  £  Z  A. j=m+2  +  g  where h ( j ) i s d e f i n e d t o be H.(x) f o r x. = j . 1  t h e summation  1  2  J  £ Z j=m+2  A. J  can be shown (.ci... Lemma 3.3.5 (3) ) t o be l e s s - t h a n o r e q u a l to  2 (P-N -2) K x )  £ S - 2h (iV Z N. . j=m+2 S-h (£)+h (£-1) 2  £[h(£) - h(£-l)]  S  2  h  3  Hence we have .. 2  £ Z A. < ,, j — j=m+l  2 (p-N -2) cb(x) c {(m+l)N ,.,h(m+l) S m+1  + £[h(£) - h(£-l)]  £ S - 2h ( j ) Z N. } j=m+2 S-h (£)+h (£-1) 2  (p-N -2) <f> (x) m _ -t2  a  _  n  d  J  A A  ^<  2  2m(p-N -2) cb(x)N ...h(m+l) m +_ m+1  2(p-N -2) cb(x)B g 2  m  =  2m(p-N -2) cb(x)N .^(m+1) m + m+1 g  ( c f . Lemma 3.3.7)  (p-N -2) <j>(x) m + xr\^ g [2B - <j>:(x)] 2  r  o  <_ 0 s i n c e 0 <j,cb(x) £ 2B. Q'.'E.D.  Remarks: (1) Note t h a t N > 1 and hence the e s t i m a t o r s \ ^ ^ dominate t h e m — • ^ m  MLE  o n l y when p i s a t l e a s t f o u r i n s t e a d o f t h r e e .  The i n c r e a s e o f t h e  dimension i s needed because we e s t i m a t e k, a p o i n t towards which t h e MLE  39  s h o u l d be s h i f t e d , by the d a t a . (2) When a l l t h e o b s e r v a t i o n s x^ o f the p P o i s s o n random v a r i a b l e s a r e e q u a l t o t h e same v a l u e , s a y X , one would c o n j e c t u r e t h a t the q  parameters identical.  o f the p random v a r i a b l e s a r e c l o s e t o one another I n t h i s case, one would e s t i m a t e  o r even  the X.'s by the grand mean  P C E x . ) / p , which i s e q u a l t o x . Our Bayesian p o i n t e s t i m a t o r s pro^i=l ° posed i n S e c t i o n 5 g i v e such an e s t i m a t e f o r t h e A^'s i n t h i s s i t u a t i o n , 1  1  as do the e s t i m a t o r s X^ ^ d e s c r i b e d i n Theorem 3.4.3 above. m  X_^  = x^ f o r a l l i i f x_^ = x^ f o r a l l i , an i n t u i t i v e l y  That i s ,  appealing  result. As Peng [1975] has n o t i c e d , i f some o f the o b s e r v a t i o n s x^,...,x^ a r e l a r g e , the unbiased w i l l be s m a l l .  estimate  o f the improvement i n r i s k , i . e . -A,  T h i s makes the s a v i n g s of t h e proposed e s t i m a t o r s  In o r d e r t o t a c k l e t h i s problem, Peng used S t e i n ' s method  small.  [1974] t o  m o d i f y h i s proposed e s t i m a t o r f o r P o i s s o n parameters t o guard a g a i n s t extreme o b s e r v a t i o n s .  However, h i s e s t i m a t o r s s t i l l  when a l l the o b s e r v a t i o n s a r e l a r g e .  give small  savings  F o r example, when the minimum o f  the A^'s i s g r e a t e r than o r e q u a l t o 12, say, the s a v i n g s from u s i n g Peng's modified  e s t i m a t o r w i l l s t i l l be s m a l l .  ^Tml  Our proposed e s t i m a t o r X  is  ~ [nil  useful i n this situation.  The s a v i n g s o f X  w i l l be much g r e a t e r  Peng's e s t i m a t o r s p r o v i d e d  t h a t the o b s e r v a t i o n s f a l l  than  into a r e l a t i v e l y  narrow i n t e r v a l , i . e . when the o b s e r v a t i o n s do n o t d i f f e r v e r y much. In g e n e r a l , the e s t i m a t o r s X  a r e expected  t o perform w e l l i n  a v e r y wide range o f parameter v a l u e s , i n c l u d i n g the cases when the A^'s a r e r e l a t i v e l y s m a l l as w e l l as when they a r e r e l a t i v e l y l a r g e .  The  * Ttnl  improvement i n r i s k o f X when p i s l a r g e and t h e \^''S  over:the MLE should be c o n s i d e r a b l e , a r e c l o s e t o one another.  especially  A simulation  40  r e s u l t .of; the b e h a y l o u r of our  conjecture.  to our  i s reported  Moreover, f u r t h e r a p p l i c a t i o n of S t e i n ' s method [1974]  estimators  A'-" "' w i l l y i e l d a more v e r s a t i l e e s t i m a t o r 1  i n t h a t , u n l i k e Peng's e s t i m a t o r , and  i t guards a g a i n s t extreme  cases i n which a l l the parameters are l a r g e or s m a l l .  i s straightforward We  now  ( c f . Peng  ^  of A  observations The a p p l i c a t i o n  d i g r e s s to d i s c u s s an obvious a p p l i c a t i o n of the techniques  d i s c r e t e exponential  families.  Q-1977], p. 18)  on  Though we w i l l not c a r r y out  b r i e f l y give::, an. example below.  Hudson 11977] b e f o r e p r o c e e d i n g  The  general the  i n t e r e s t e d reader  should  read  to the f o l l o w i n g example because we  the n o t a t i o n employed by Hudson.  Using  h i s n o t a t i o n , we  will  define  r  CD  m = Min {x.;}. and recall t(x.) = t. i f x„ = j 1=1  1  1  J  1  x.^m x (2)  b  v  x  i  = 1/t.  +  x  Z  ™ -  k=2  if x  k+m  c  = m+2,,,..  1  = l/t  i f x. =  =0  i f x. = m  ' 1n  x  m+1  x  (3)  S=  i  f  1=1  (4)  , p .=  ( -N -3)  m  P  m  +  •  X  g.(x) = -r  p  • b  x  i  , g(x),=-(g Cx).,....,g ( X ) ) . 1  p  Then i f p >^ 5, the estimator T + g(x) dominates T = (tCX^',... ,t(X )) under squared error loss. 2  <p  P  :".  Csince i t i s o u t s i d e the scope of t h i s s t u d y ) ,  d e t a i l e d a n a l y s i s here  use  A  [1975]).  i n t h i s s e c t i o n to extend Hudson's r e s u l t  we  i n S e c t i o n 7, which s u p p o r t s  N  The improvement in risk exceeds -L-I  b  t _,, t.,  p  2  o  41  SECTION 4.  4.1  ESTIMATION UNDER K-NSEL  Introduction Theorem 2.2.9 p r o v i d e s us w i t h a c l a s s of e s t i m a t o r s  the MLE, X, under n o r m a l i z e d  squared  Pe r r o r l o s s L^ = E i=l  dominating 2  (A  - X^) /X^.  S i n c e X i s minimax under L^, t h e e s t i m a t o r s g i v e n i n Theorem 2.2.9 a r e a c t u a l l y minimax e s t i m a t o r s o f X.  One way t o show t h a t an e s t i m a t o r o f  the form X + f ( X ) i s minimax i s to show t h a t f s a t i s f i e s  the c o n d i t i o n s  i n Lemma 2.2.6 and t h a t 2 p f.(x+e.) A.. = E +2 1 . i X.+l X=l X 1  p E [f.(x+e.) - f (x) ] X X X x=l  1  <_ 0 f o r a l l x e J  .  +  T  ( C o r o l l a r y 2.2.8).  In t h e f i r s t p a r t o f t h i s  s e c t i o n , we s h a l l use C o r o l l a r y 2.2.8 to  show t h a t the c l a s s o f minimax e s t i m a t o r s g i v e n i n Theorem 2.2.9 c a n be enlarged  i n two ways.  The remainder o f the s e c t i o n i s devoted t o attempts  to f i n d e s t i m a t o r s b e t t e r than the MLE under t h e k-normalized error loss  (k-NSEL) f u n c t i o n L (X,X) 1C  =  P E  squared  A k (X. - X.) /X. , w i t h k >_ 2. . - . 1 1 1 2  1=1  We s h a l l make u s e o f Lemma 2.2.7 to prove the main r e s u l t s and s h a l l a l s o d i s c u s s the m o t i v a t i o n f o r u s i n g k-NSEL. 4.2  Minimax E s t i m a t o r s We s h a l l show i n t h i s  s u b s e c t i o n t h a t a l a r g e c l a s s o f minimax  e s t i m a t o r s can be o b t a i n e d which i n c l u d e s t h e c l a s s o f e s t i m a t o r s proposed by Clevenson and Zidek  [1975] as a s u b c l a s s .  The e s t i m a t o r s we c o n s i d e r  a r e o f the form X + f ( X ) and l o s s f u n c t i o n L ^ i s used. 4.1,  we need o n l y show t h a t A^(x)  +P <_ 0 f o r a l l x e J  As remarked i n  42  ij>(z) be a n o n i n c r e a s i n g r e a l - v a l u e d f u n c t i o n such t h a t z + i|>(z)  Let is  for z e J .  nondecreasing  The f o l l o w i n g theorem g i v e s a c l a s s o f  +  e s t i m a t o r s which dominate the MLE X and hence a r e minimax.  Theorem  4.2.1. independent  Suppose X ^  ^  P o i s s o n ( A ^ ) , i = l , . . . , p , p J^2.  D e f i n e A ( X ) = [1 - cb ( Z ) / ( Z + I ( J ( Z ) ) ] X where  (1)  i|j(z) >_ b > 0 f o r some b  (2)  cb(z) i s n o n d e c r e a s i n g  (3)  0 £<|>(z+l) <_ 2Min{(p-l) ,^(z)} f o r a l l z e . J  Then f o r a l l A = (A. 1 squared  .  A ) , A dominates the MLE X under the n o r m a l i z e d p  e r r o r l o s s f u n c t i o n L^.  Proof:  -<K'z)x  Let  f . ( x ) = -—, ,,, . l z+nz)  ifx e  = 0  P J  +  otherwise.  We need to show t h a t 2 p f (x+e ) A, = E +2 1 . , x.+l i=l l ;  In  p £ [f.(x+e.) - f . ( x ) ] < 0. . 1 1 l — i=l  fact, (b (z+l)(z+p) 2  =  1 "  [  z  +  i  +  l  K  z  +  i ) ]  cb(z+l)  - [z+l+iKz+D]  2  _ "  r  cb(z+l)(z+p) _  L Z  +1+*( +D Z  , (z+p)cb(z+l) _  z+l+ij/(z+l)  ( s i n c e z+^(z) i s n o n - d e c r e a s i n g  "  cb(z)  Zl  J  (z+p)(z+i|;(z))-z(z+l+^(z+l))  z+ij;(z) and cb(z) >_ 0)  1  43  <p(z+l) (z+p)(p(z+l) _ - [z+l+i|;(z+l)] z+l+iKz+1)  =  Note t h a t  z]&(z)-ifr (z+1) ]+(p-l) z+pifr(z) ,  r  iKz) - ^(z+1)  "  z+i|j(z)  -  >_ 0 by assumption and t h a t z+\\)(z) i s non-  d e c r e a s i n g , so we have  <  <Kz+l) [z+l+^(z+l)] Kz+1)  {(  _  z + p  )  ( | )  { z [ ( p ( z + 1 )  (  z +  i) .  2  ((p-l)z + piKzX)}  _ 2 ( p - l ) ] + [<p(z+l)-2^(z)]} P  fz+l+^(z+l)] 1 0 by c o n d i t i o n (3) o f the theorem. Therefore  A = (1  - "^^y)'  x  dominates X. Q.E.D.  ty(z)  The s p e c i a l case where  = b > 0 i s s t a t e d as a c o r o l l a r y below.  C o r o l l a r y 4.2.'2. ' _= independent ^ . Suppose X^ ^ Poisson r  ' "  -' • '  R  "  ^  A  f  .i . . (A£); I = l , ^ . J p ,  p >_ 21  *7 \  Then the e s t i m a t o r A = X - „,, • -X of -z+b'.. J l o s s f u n c t i o n L^ where  A dominates the MLE X under the  (1) :b > 0 (2)  cp(z)  i s nondecreasing  (3) .0 <_ <p(z)  i Min{2(p-1) ,2b} and <p(z)  t 0.  Note t h a t the constant b g i v e n i n the c o r o l l a r y i s an a r b i t r a r y p o s i t i v e r e a l number, w h i l e the e s t i m a t o r s g i v e n i n Theorem 2.2.9 r e q u i r e b >_ p-1.  Moreover, the theorem r e q u i r e s t h a t 0 <^ <p(z)  <^ 2(p-1), and hence  the c l a s s o f e s t i m a t o r s g i v e n i n Theorem 2.2.9 i s i n f a c t a s u b c l a s s o f ours.  44  -The e s t i m a t o r X s h r i n k s the MLE towards the o r i g i n by the amount . *.u  For every b, the maximum s h r i n k a g e a l l o w a b l e i f X i s t o dominate  MTT? •  the MLE i s  Min{2(p-1) ,2b}x  z+b  '  w  ( c o o r d i n a t e - w i s e ) whenever b > p-1.  r,. • . .  n  l  c  n  1  S a  •  •  *  c -u  i n c r e a s i n g f u n c t i o n of b  n  0 < b <_ p-1 and a d e c r e a s i n g f u n c t i o n whenever  T h e r e f o r e the maximum s h r i n k a g e  An a p p l i c a t i o n of C o r o l l a r y 4.2.2 i n t e r e s t i n g e s t i m a t o r s o f X.  i s o b t a i n e d when b = p-1.  above g i v e s us s t i l l more  The r e s u l t i s s t a t e d i n C o r o l l a r y  4.2.3  below. C o r o l l a r y 4.2.3.  Suppose  independent ^ Poisson  (X^) , i = l , . . . , p , p ^ 2.  Then the  estimator A = (1 - " ^ - ) X  of X  t  dominates the MLE X under the l o s s f u n c t i o n L ^ where (1) t >_ 1, c > 0 (2)  0 < a < Min { 1 ^ = 1 ) .  ,  C  ,  ^ }  .  Proof: Rewrite X = {1~  X as f o l l o w s : [1 -  (^|ga-)t]}X  where 0(z) = ( ^ + c ) - ( + c - a ) t  t  2  (z+c)  t _ 1  We s h a l l show t h a t ( i ) 0(z) i s non-decreasing  and  ( i i ) .0 1.0 (z) .<_ Min{2(p-1) , 2c}. .Now,'the d e r i v a t i v e o f 0 (z) i s 9'(z) = 1  (z+c-a)  t  1  (z+c-a+ta)  (z+c)*  45  S i n c e (z+c)*" >_ (z+c-a)*" + t a ^ + c - a ) * " Hence 0(z) i s n o n - d e c r e a s i n g . z >_ 0.  f o r a l l z >_ 0, we have 0'(z) >^ 0.  Next, i t i s c l e a r t h a t 0 <_ 6(z) f o r a l l  Now  0(z) • t a C z + c - a ) ^ (z+cK  1 +  1  where i K z ) i s such t h a t l i m i|)(z) = 0 .  As a r e s u l t , 0(z) £ l i m 0(z) = t a .  From c o n d i t i o n ( 2 ) , we have 0(z) £ m i n { 2 ( p - l ) , 4.2.2  we  t c , 2c}.  By  Corollary  see t h a t the e s t i m a t o r  dominates the MLE  under the l o s s f u n c t i o n L^. Q.E.D. 2  The e s t i m a t o r X(X) = (1 - (p-1)/(Z+p-1))  X, which i s an e s t i m a t o r  d e s c r i b e d i n the p r e v i o u s c o r o l l a r y w i t h t = 2 and a = c = p-1, s h r i n k s more towards the o r i g i n than does the e s t i m a t o r X  = (1 - (p-1) / (Z+p-1) )'X.  ^* Thus, A should g i v e a b e t t e r e s t i m a t e .of X than  X^,  i = l , . . . , p a r e c l o s e enough to z e r o .  us an i n t e r e s t i n g i n s i g h t as to why  i f the  parameters  The f o l l o w i n g argument g i v e s  we might a r r i v e a t e s t i m a t o r s of the  X = (1 - ^ r - ) X . Z+a  form  2  L e t X^ ^ P o i s s o n (X ), Y^ =  X  2/xT,  9^ =  Y_^ ^ N(0^,1),  2\(xT,  i = l , . . . , p , be m u t u a l l y independent  i = l,...,p.  i = l , . . . , p , and  and l e t  I t i s a p p r o x i m a t e l y t r u e t h a t 7.  t h a t the Y^'s  are mutually  independent.  That i s , a p p r o x i m a t e l y , Y = (Y.,...,Y ) ^ N (0,1 ) , where I i s the p x p 1 p p p p i d e n t i t y m a t r i x and 0 = (0^,...,6 ) . The James-Stein e s t i m a t o r 0 = (0^ 0p) of 0 under squared e r r o r l o s s i s 0^ = (1 - r/Y'Y)Y^, ^ 2_f 2. c — A  A.  1 = l,...,p.  A,  A  Or, i n terms of X and  X,  X.  1  = (1  - —)*/xT, Ci  W.  i = l,...,p,  46  where Z =  E  X . , c = r/4,  i=l We  and  p r o b a b i l i t y of b e i n g  A =  z e r o , we  a i s a p o s i t i v e r e a l number. ( 1- ^ - ) Z+a One  (A.,...,A 1  thus have the e s t i m a t o r  A =  A =  1  2  X  ) i s an e s t i m a t o r  of A .  p  ( 1 - c/Z)  X of A .  Since  Z has  a positive  a r e prompted to r e p l a c e Z by Z + a, where We  thus arrive.-at'.the  estimator  of A .  might argue t h a t the above r e s u l t i s o b t a i n e d  through the  of squared e r r o r l o s s i n s t e a d of n o r m a l i z e d squared e r r o r l o s s . n o t i c e t h a t the squared e r r o r l o s s f u n c t i o n i s a p p l i e d o n l y i n  use  However, estimating  2 a s y m p t o t i c a l l y , (6L - 6^) is O ( A ^ ) , i = l,...,p (i.e. 2 (0. - 0.) / A . < °°). The n o r m a l i z e d squared e r r o r l o s s f u n c t i o n  8, and lim  X -*»  i trie same asymptotic p r o p e r t y , Hence, our  r e s u l t can be  In order  "  2  i . e . ( A ^ - A ^ ) /X  = 0(A_^), i =  thought of as i f i t i s o b t a i n e d  that estimators  of the form A = X -  not a n e c e s s a r y c o n d i t i o n .  and  A will still  In f a c t ,  dominate the MLE  X.  <p(z)  The  l,...,p.  under  can be d e c r e a s i n g  independent A* Poisson  v..  ( A ^ ) , i = l , . . . , p (p  <!>(z)  A  t  Let A =  ( 1  ZZT)X (z+b)  be an e s t i m a t o r  of A where  t + 1  (1)  t >_ 0 , b > t + 1  (2)  <f> (z) i s n o n d e c r e a s i n g i n z  (3)  tp (z) > 0 and  (4)  <*V(z) — ^ — < Min{2(p-t-l),2(b^t-l)}.  t  t  u+br  cp^z)  f  0  for a l l z  f o l l o w i n g theorem s t a t e s t h i s ..'  4.2.4  Suppose X ^  MLE  nondecreasing  fact. Theorem  L^.  V,, X dominate the Z+b  X under n o r m a l i z e d squared l o s s , the requirement t h a t ep(z) be is  has  1  ^2).  47  Then f o r a l l A = (-A,,...,A ) , A has s m a l l e r r i s k than A° = X when the 1 p l o s s f u n c t i o n i s g i v e n by L^. Proof: -cb (z)x —— (z+b) t  L e t f . (x) = 1  p i fx e J  i  +  t+J  =0  otherwise, i = l , . . . , p .  A g a i n , we need t o show  2 A  p Z  =  f (x+e ) + 2 1  p Z  1  1  X=l  [f.(x+e.) - f . ( x ) ] < 0.  1=1  Now <|> (z+l) (z+p)  -eb (z+l)(z+p)  2  A  i  =  21+2"  —  (z+b+l)  1  +  2  E+i: (z+b+1)  [  Z t + Z  cb (z+l)(z+p)  +l (z+b) ]  t  t + 1  _  2  < —/ - u o - -*ZTT-+ ^  4> (z+D [ L itH-1+• • .,xt+l1 ^ ^-/ L , (z+b+1) (z+b) U + P  2  2 t + 2  (z+b+1)  (since $  cb..(z)z  +  T  i s non-decreasing) cb (z+l)(z+p)  1  =  V ± z  1 ) {  -  C  •  7  -2t+2  +  2  (  z  +  p  )  (z*b+I)  [  t+i " (z+b)  Observe t h a t t h e f u n c t i o n f d e f i n e d by f ( z ) =  L  -E+T  ^t+T  ]  (z+b+1)  _ is strictly  (z+b) Hence f ( z + l ) - f ( z ) >^ f ' ( z ) . Consequently t + i  convex f o r z >_ 0.  1  C  (z+b+1)  C  (z+b)^  (z+b)  C + Z  (z+b)  ]  (z+b)  t + J :  C  1  48  <  0, 4> (z+l) t  since  <_ 2 M i n { p - t - l , b - t - l } ,  i . e . condition  C4) o f our  (z+b+1) theorem. Q.E.D. I f b=p-l, the theorem h o l d s even when c o n d i t i o n i s replaced  (4)'  (4) i n the theorem  by  (z+b)  _ t  4> (z) < 2 ( p - f - l ) . t  That i s , (z+b) *" 4> (z) has a l a r g e r upper bound. t  We  state this result  i n the theorem below. Theorem 4.2.5. L e t X, A and A be as g i v e n  i n Theorem 4 w i t h b=p-l.  Suppose <f> (z) s a t i s f i e s the f o l l o w i n g t  (1)  t >/0,  (2)  cb^z) i s n o n - d e c r e a s i n g i n z  (3)  cb (z) > 0 and <j> (z) 4 0  conditions:  p-1 > t  t  t  Then f o r a l l A, A has s m a l l e r L  r i s k than X when the e r r o r l o s s f u n c t i o n i s g i v e n  by  r  Proof: The p r o o f  here i s e s s e n t i a l l y the same as t h a t of Theorem 4.2.4.  can be shown t h a t  It  49  A  i  ±  2t+i  +  2  *  (2+ P ) Z t + 1  — F  t  (Z+p-1)  t  ^.T±31+I +  2  t t ^ D t  (-z+p)  -  - — ~T i  (Z+P)  C  *  <P^(z+l)  r  r+r  +2  M >  (Z+p-1)  ]  t+1  —f ^]  -  (z+p-1)  ^7+T  )  8  (z+p-1)  2+1  — L i : (z+p) ^ ( z + p - D " (z+p-D «> (z+1) <?. (z+1) ' (z+p-D ) - 2(p-t-l)]; [ (—z + — p— t + i  t  t + 1  1  t + 1  <_ 0 (by c o n d i t i o n  (4) ') .  The second i n e q u a l i t y h o l d s s i n c e the f u n c t i o n f ( z ) = — — — (z+p-1) strictly  is  convex f o r z >_ 0. Q.E.D.  When t = 0, the above theorem i s Theor em 2.2.9 of S e c t i o n 2.  The  theorem, due t o Clevenson and Z i d e k [1975], suggests t h a t e s t i m a t o r s of ^ 6 (Z") the form X = (1 - _, \)X Z+p-1  w i l l dominate the MLE X p r o v i d e d t h a t  0 <_ 8(z) <_ 2(p-1) and t h a t 8(z) i s n o n d e c r e a s i n g . from the theorem above, the requirement not n e c e s s a r y .  However, as we see  t h a t 6(z) be n o n d e c r e a s i n g i s  A s i m p l e example i s g i v e n below.  Example 4.2.6. Let  X = (1 -  C  (Z+p-1)  )X w i t h 0 ^ f < p-1 and 0 < c < 2 ( p - t - 1 ) . t + ±  Then by our theorem, X i s b e t t e r than X under the n o r m a l i z e d squared loss function. -0(z) =  We can w r i t e X i n the form  error  0 (Z) (1 - ^ ^)X, where  ^-'j and note t h a t 0(z) i s s t r i c t l y d e c r e a s i n g f o r a l l z >_ 0. (z+p-1)  50  4 ."3  B e t t e r E s t i m a t o r s Under L, k The usage o f unbounded l o s s f u n c t i o n s has been a c o n t r o v e r s i a l t o p i c  because o f the famous S t . P e t e r s b e r g ' s c o n s i d e r bounded l o s s f u n c t i o n s .  paradox; hence i t i s n a t u r a l t o  Noting  t h a t the l o s s f u n c t i o n  is  unbounded both when the ^ ' s a r e s m a l l and when they a r e l a r g e , we a r e A  P  prompted t o c o n s i d e r the l o s s f u n c t i o n s L (A,A) = (k 2l 2) which a r e bounded when the A^'s a r e l a r g e .  E x=l  " 2 k (A. - X.) /A.  This i s desirable i f  we a r e p r i m a r i l y i n t e r e s t e d i n the problem o f e s t i m a t i n g A when the A^'s are r e l a t i v e l y l a r g e .  Furthermore, when the A^'s a r e l a r g e , the v a r i a n c e s  of the d i s t r i b u t i o n s a r e l a r g e , and the e s t i m a t i o n problem i s more d i f - - . ficult.  I n t u i t i v e l y we might expect the MLE t o perform f a i r l y w e l l when  the A.'s a r e r e l a t i v e l y s m a l l , s i n c e i n t h i s case, d i s t r i b u t i o n s a r e s m a l l , and the o b s e r v a t i o n s fall  are exceedingly  c l o s e to the mode (hence the range o f the o b s e r v a t i o n s  likely and  the v a r i a n c e s o f the  small).  l i k e l y to  i s very  However, when the A^'s a r e l a r g e , the v a r i a n c e s a r e l a r g e  the o b s e r v a t i o n s  a r e s c a t t e r e d a c r o s s a wide range o f v a l u e s .  This  l e a d s us t o c o n j e c t u r e t h a t i t i s p o s s i b l e t o improve on the MLE when the A_/s  are large. There i s another reason why we might t h i n k o f u s i n g  f u n c t i o n i n the e s t i m a t i o n problem.  as our l o s s  The parameter o f a P o i s s o n d i s t r l b u ^ •.  t i o n can be thought o f n o t o n l y as the mean o f the d i s t r i b u t i o n , b u t a l s o as the v a r i a n c e . With t h i s i n mind, a n a t u r a l l o s s f u n c t i o n t o use would .P r. O *• be E ..(1> A./A.) , which i s L.(A,A) (see Brown [1968]). 1=1 - 1 1 Before  s t a t i n g the next theorem, we i n t r o d u c e some n o t a t i o n t o be  used i n the theorem.  51  Definitions: S =  (1)  (2)  (kl (X.+k) =  F  E 1=1  v  1  Z 1=1  S =  ;  (x.+k)  =  w  1  (3)  S  1  = S - (X + k )  (4)  S  1  = S - (x.+k)  Theorem 4.3.1  H  E 1=1  (X.+l)---(X.+k) 1  Z 1=1  X  (x.+l)---(x.+k) 1  X  ( k )  (k)  below g i v e s us a c l a s s of e s t i m a t o r s A =  u n i f o r m l y dominating  the MLE  under the l o s s f u n c t i o n L ^  (A^,...,A )  (k >_ 2).  These  e s t i m a t o r s have the f o l l o w i n g p r o p e r t i e s : th (1)  I f the o b s e r v a t i o n from the i  population i s small  than k ) , then the e s t i m a t o r A_^ of A_^ i s the same as the (2)  (less  MLE.  I f the o b s e r v a t i o n i s l a r g e ( g r e a t e r than or e q u a l to k ) ,  the e s t i m a t o r X. Using  of X.  s h r i n k s the MLE  then  towards z e r o .  the d e f i n i t i o n s g i v e n above, the theorem and  i t s proof  are  , i = l , , . . , p , p >_ 2,' .and  the  s t a t e d as f o l l o w s : Theorem 4.3.1. independent Suppose X^  ^  Poisson  l o s s f u n c t i o n i s L^XA^A).  (X/)  Then the e s t i m a t o r A g i v e n below dominates  the MLE X u n i f o r m l y i n A = CA-p • . •,A " th ^' A. = i c o o r d i n a t e of A  ) :  I  <j>(Z) X. ( X . - l ) • • • (X.-k+l) = X_. 1  1  1  1  S "+ X ( X - 1 ) . • • (X^.-k+l) 1  ±  i  52  where (1)  P E  z =  x.  i=l  1  (2)  <|>(z) i s a r e a l v a l u e d f u n c t i o n i n c r e a s i n g i n z  (3)  0 ± <|>(z) 1 2k(p-1) and <|>(z) 2 0.  Proof: -<f>(z) x. ( x . - l ) • • • (x.-k+l) L e t f . (x) = — : S + x (x - l ) " - ( x «k+l) 1  i f x. >_ 0, i = 1,. ..,p  1  =0  1  i f x. < 0. I  We see t h a t t h e f ^ ' s s a t i s f y t h e c o n d i t i o n s g i v e n i n Lemma 2.2.6.  Hence  Lemma 2.2.7 g i v e s us an unbiased e s t i m a t e A^ o f t h e d e t e r i o r a t i o n i n r i s k of A. We have D, = E A  f (x+ke.)  P  a  n  d  A  k  I  =  = R(A,A) - R(A,X)  2  - — ( k i  1=1 ( x . + k ) l  k  f.(x+ke.)  P  +  2  W  ^ i 1=1 (  x  +  k  1  )  ——  1  - f.(x+(k-l)e.)  —m  (x.+k) l  ~  W  S u b s t i t u t i n g t h e f 7 s i n the f o r m u l a , we have ^2. . . . M | k  P  +  k  I  +  2  -<j>(z+k) • (x.+k) +  i=i  s  <  l  S  S  s ^ x . + k - i ) ^  s  ,2. . . / ^ (|) (z+k) , 2(t(z+k)  <b(z+k-l)x.  -(x,+k)[S - k ( x + k - l ) i  P  ( k _ 1 )  J  v  v  1=1  S  1  + (x k - l )  4  (  k  )  +  ( s i n c e <j>(z) i s i n c r e a s i n g and <f>(z) ^_ 0) 12 v - v (j) (z+k)  p +  2cj) (z+k)  S  S  Now observe t h a t S > S and  £  -kS +  1=1 S  1  1  k(x.+k)  + (x + k - l )  + (x^+k-l)  (k) S > (x^k)  ( k )  I  (k)  ( k )  ] •+ x S  53  Hence, we  1  have,  0  ( s i n c e 0 <_ <p(z+k) <_ 2 k ( p - l ) and  <J>(z+k) i  0) . Q.E.D.  Remarks: (1)  When k = l , Theorem 4.3.1  i n Clevenson and (2) • I f x no s h r i n k a g e  Zidek < k-1,  of the MLE  the l i k e l i h o o d  i s the same as Theorem 2.2.9, a r e s u l t  [1975]. then  g i v e s the same e s t i m a t e  takes p l a c e .  We  as the MLE,  i.e.  see from t h i s t h a t as k  <?,  t h a t s h r i n k a g e w i l l be i n d i c a t e d becomes s m a l l e r and  I t i s i n t u i t i v e l y c l e a r t h a t i f we s m a l l o b s e r v a t i o n s l e s s than we  s h r i n k the MLE,  we  shrink:large observations.  p a r t l y because when the o b s e r v a t i o n s  should  smaller.  shrink  This i s  are s m a l l , the u n d e r l y i n g parameters  a r e l i k e l y to be s m a l l , which means t h a t the v a r i a n c e s a r e s m a l l and  hence  each o b s e r v a t i o n i s l i k e l y to be c l o s e to the v a l u e of i t s r e s p e c t i v e parameter.  On  the o t h e r hand, l a r g e r v a l u e s of the parameters  to l a r g e r v a r i a n c e s of the random v a r i a b l e s , and s c a t t e r i n g of the o b s e r v a t i o n s l e a d s one MLE  the h i g h e r p r o b a b i l i t y of  to s u s p e c t  t h a t i n t h i s case  i s l e s s r e l i a b l e than when the o b s e r v a t i o n s a r e s m a l l ; one  t h e r e f o r e be more i n c l i n e d  to a d j u s t the MLE.  correspond  The  the  might  i n t e g e r k i n the  loss  f u n c t i o n L ^ r e f l e c t s the degree of concern  about m i s e s t i m a t i o n  r e l a t i v e l y s m a l l A^'s.  r e s u l t i n l o s s f u n c t i o n s which  Large v a l u e s of k  are v e r y s e n s i t i v e to changes i n the A^'s. those o b s e r v a t i o n s  of  S i n c e our e s t i m a t o r s  t h a t a r e g r e a t e r than or e q u a l to k but l e a v e  shrink the  54  o t h e r s untouched, the i n t e g e r k can be i n t e r p r e t e d as an i n d i c a t o r o f a person's w i l l i n g n e s s t o move the MLE f o r b e t t e r e s t i m a t i o n r e s u l t s . Choice o f a s m a l l v a l u e o f k would p r o b a b l y  r e s u l t from a p r i o r  t h a t the A^'s a r e s m a l l , and hence the person  i s willing  belief  t o move even  s m a l l o b s e r v a t i o n s which, as p o i n t e d o u t above, a r e supposed to be more reliable.  Choice  prior belief  o f a l a r g e v a l u e o f k would p r o b a b l y  t h a t the A^'s a r e n o t s m a l l , and thus the person  not t o move the o b s e r v a t i o n s (3)  r e s u l t from a i s inclined  i f they a r e n o t l a r g e enough.  Theorem 3.1 o f Clevenson and Zidek  [1975] suggests  t h a t e s t i - '_  mators A = (1 - <p (Z) / (Z+p-1) )X o f X s t i l l dominate the MLE under a g e n e r a l l o s s f u n c t i o n L (A,A) = K.  P Z  . i 1  nonincreasing function.  - 2 K(A.) (A. - A.) /A. where K > 0 i s some  1  1  1  1  k-1 When K(y) = 1/y , L =  1  i s the k-NSEL L. .  However, our e s t i m a t o r s do n o t s h r i n k o b s e r v a t i o n s  t h a t a r e l e s s than k;  o n l y those o b s e r v a t i o n s g r e a t e r than or e q u a l t o k a r e moved. if  A^ >^ k, our e s t i m a t o r s guard a g a i n s t unnecessary s h r i n k a g e  o b s e r v a t i o n s happen  to be s m a l l ( i . e . < k ) .  S i n c e the  Therefore i f the  Clevenson-Zidek  e s t i m a t o r s h r i n k s a l l non-zero o b s e r v a t i o n s , we a r e l e d t o c o n j e c t u r e t h a t our e s t i m a t o r s a r e b e t t e r than t h e i r s . i n terms o f the p e r c e n t a g e i n s a v i n g s compared t o the MLE when the A^'s a r e r e l a t i v e l y l a r g e ( i . e . when Min{A_^}^_ k^_ 2 ) .  Some s i m u l a t i o n r e s u l t s which support  this  conjecture  a r e r e p o r t e d i n S e c t i o n 7. The next when k-NSEL L  theorem i s a g e n e r a l i z a t i o n o f C o r o l l a r y 4.2.2 t o the case i s used, where k i s any p o s i t i v e i n t e g e r .  Theorem 4.3.2. Suppose X = (X^,...,Xp)  i s as g i v e n i n Theorem 4.3.1  and  X = (X,,...,A ) i s an e s t i m a t o r o f X. 1 p  <Kz)x.  Let X  r- T T T S^X.^+b l  = X 1  1  where (1)  ( k )  , i=l,...,p,  k i s .a p o s i t i v e i n t e g e r  (2)  <j) i s non-decreasing  (3)  0 < <f>(z) £ min{2  (4)  b > -  (  b  |^~^  k  , 2k(p-l)}  !  (p-l)[k!].  Then f o r a l l X, X dominates X under the. e r r o r l o s s f u n c t i o n L, k Proof: -<j>(z)x f . (x) = — rr-r 0  Define  p ifx e J  0  sW -H, k>  l  = 0  otherwise,  i-1,...,p.  We need to show t h a t p f.(x+ke.) S — pfr- + 2 1-1 ( x . k )  A, = k  p E 1=1  ( k )  +  f.(x+ke.)-f.(x+(k-l)e.) (x.+k) — — 1—?-: (x. k) 1  ( k )  +  < 0.  I  n  d  e  e  d  * k  1 2 / J_T_\c i r ,i \ < * < > - 2 z + k  ~  [S+b]  S  2  S  +  b  p \  kS-k(x.+k) -. i  ( k )  +kb  i=lS (x. k-l) Vb 1  (x  +  +  56  ±  [  l  !  2  ^  _ liCz+ki  [S+b]  (since  <. $ (z+hl [s+b]  (  k  (  p  _  1  )  s  +  p  k  b  )  [S+b]  kS  - k(x +k) i  ( k )  +kb  > 0)  [S((f,(z+k)-2k(p-l)) - 2pkb]. z  Now observe t h a t S(<J)(z+k) - 2 k ( p - l ) )  —2pkb  <_ pk! (<j>(z+k) - 2 k ( p - l ) ) -2pkb ( s i n c e c o n d i t i o n (3) g i v e s <|>(z) £ 2 k ( p - l ) )  -pk!  [•(*«  -2  ]  < 0 by c o n d i t i o n (3) . Hence A. < 0. k — Q.E.D.  57  SECTION 5.  5.1  BAYESIAN ANALYSIS  Introduction In t h i s s e c t i o n , we  e x p l o r e the problem o f e s t i m a t i n g the P o i s s o n  parameters from another p e r s p e c t i v e . i  R e c a l l that  ^ Poisson (A^),  = l , . . . , p , and t h a t the X^'s a r e m u t u a l l y independent.  Moreover,  o n l y one o b s e r v a t i o n i s taken from each of the p P o i s s o n random v a r i a b l e s . In c o n t r a s t to the f r e q u e n t i s t approach taken i n the l a s t s e c t i o n , s h a l l take a B a y e s i a n approach and show how the MLE  X = (X^,...,X^).  we  to f i n d e s t i m a t o r s b e t t e r than  Observe t h a t X i s a Bayes e s t i m a t o r i f p r i o r  knowledge of the parameters A^ i s n o n - i n f o r m a t i v e (vague) and the A^'s a r e independent.  When s u b s t a n t i a l p r i o r knowledge i s a v a i l a b l e ,  sign-  i f i c a n t improvement on the u s u a l e s t i m a t i o n of p r o c e d u r e would be expected by means of B a y e s i a n methods.  That i s , i n c e r t a i n s i t u a t i o n s , we  can  i n c o r p o r a t e the i n f o r m a t i o n a t hand about the p r i o r d i s t r i b u t i o n i n a B a y e s i a n manner and o b t a i n e s t i m a t o r s of A = (A^,...',A  ) b e t t e r than X.  R e c e n t l y , some attempts have been made t o study the problem o f s i m u l t a n e o u s l y e s t i m a t i n g the parameters o f s e v e r a l independent P o i s s o n random v a r i a b l e s from the B a y e s i a n p o i n t o f view.  As mentioned  previously,  Clevenson and Z i d e k [1975] propose a c l a s s o f e s t i m a t o r s t h a t dominate MLE u n i f o r m l y under n o r m a l i z e d squared e r r o r l o s s . Bayesian i n t e r p r e t a t i o n of t h e i r r e s u l t s .  the  They a l s o p r o v i d e a  Leonard 11972] assumes  t h a t the A^'s a r e independent and i d e n t i c a l l y d i s t r i b u t e d , w i t h 2 2 I n A^ ^ N(y,a ).., i = l , . . . , p f o r g i v e n y and a , t h a t y i s u n i f o r m l y 2 d i s t r i b u t e d over the r e a l l i n e , and t h a t vn/a  i s independent o f y and  has a c h i ^ s q u a r e d i s t r i b u t i o n w i t h v degrees of freedom.  Modal e s t i m a t e s  58  of I n A^, i = l , . . . , p a r e proposed.  L a t e r , i n another paper,  [1976] b r i e f l y d i s c u s s e s the problem a g a i n . meters A ^ a r e exchangeable the p r i o r d i s t r i b u t i o n s  Leonard  He assumes t h a t the p a r a - .':-  i n the sense o f de F i n e t t i  [1964].  That i s ,  (two a t a time, t h r e e a t a time, e t c . ) o f the  A ^ ' s a r e i n v a r i a n t under permutation o f the s u f f i x e s . g > 0, the A ^ ' s a r e assumed t o be independent  Given a > 0 and  and each A ^ has gamma  density  n(X  |.a,3) = B A a  a i  _  1  e"  e  A  i  /r(a)  = 0  forX  > 0  otherwise.  In the second stage o f the d i s t r i b u t i o n , I n 3 i s assumed to be u n i f o r m l y d i s t r i b u t e d over the r e a l l i n e and no p r i o r d i s t r i b u t i o n o f a i s suggested. In t h i s s e c t i o n , we g e n e r a l i z e the above r e s u l t s by a d o p t i n g v a r i o u s p r i o r d i s t r i b u t i o n s on a, a l l i n c l u d e d w i t h i n the broad f a m i l y o f g e n e r a l i z e d hypergeometric  f u n c t i o n s , and develop the j o i n t  posterior  d i s t r i b u t i o n of A as w e l l as the m a r g i n a l p o s t e r i o r d i s t r i b u t i o n s o f the In A ^ , 1=1,...,p. w i t h the MLE.  P o i n t e s t i m a t o r s o f the A ^ ' s a r e proposed and compared  By means o f a computer s i m u l a t i o n which w i l l be r e p o r t e d i n  S e c t i o n 7, i t i s found t h a t i n c e r t a i n s i t u a t i o n s , e s p e c i a l l y when the parameters  A ^ a r e c l o s e t o each o t h e r and the l o s s i s squared e r r o r , a  s u b s t a n t i a l s a v i n g s over the MLE w i l l r e s u l t . B a y e s i a n s o l u t i o n f o r our problem, d e n s i t y of A...  We s h a l l f i r s t d e r i v e a  and then develop the m a r g i n a l p o s t e r i o r  59  5.2  E s t i m a t e s of the Parameters The  following  d i s t r i b u t i o n s are relevant  to a B a y e s i a n s o l u t i o n  of  our problem: (1)  Given A^,  i=l,...,p,  the o b s e r v a t i o n s x^,...,x^ a r e  and have P o i s s o n d i s t r i b u t i o n s w i t h parameters A^,...,A The j o i n t p r o b a b i l i t y mass f u n c t i o n  independent  respectively.  of x g i v e n X i s  -X. x. P e X. f(x|X) = n — r ^ -. i-1 i x  (2)  The  A_/s  a r e exchangeable a p r i o r i ,  i s d e s c r i b e d i n two (i) density,  !  s t a g e s as  G i v e n a and  i . e . f o r a > 0,  and  follows:  B, the X^ a r e independent  /T(a)  e  ifX  X  = 0  density,  :;a and  g a r e independent,  i . e . , the d e n s i t y  ±  > 0  otherwise.  and  g has a n o n - i n f o r m a t i v e  of g i s p r o p o r t i o n a l  a i s assumed to have the t r a n s l a t e d mass  and have a gamma  g > 0,  ir(A |a,8) = A ?  (ii)  the p r i o r d i s t r i b u t i o n  geometric  to 1/g.  The  parameter  d i s t r i b u t i o n with  probability  function  (1-y)  if  a  = 1  a-1 (l-y)y  i f a = 2,3,..., where 0 <_ y < . l .  The hyperparameter y i s a s s e s s e d a  priori.  prior  60  -The j o i n t d e n s i t y of x, A, a, B g i v e n y i s p r o p o r t i o n a l to -A. xv 1,  S  6  * i=l  i  i  A  5  ,  -BA.  „ot,ot-l  I  A  * . . i=l  " i  n  i  6  1 all " * B *  T(a)  y  C o n d i t i o n i n g on x and i n t e g r a t i n g w i t h p o s t e r i o r d e n s i t y o f A and ct, /, I •> -A f ( A , a x,y) « e £  '  respect  to B g i v e s the j o i n t  g i v e n x and y,  p x p . / • \• •„ i ,ot-l T(pa) II ..A. [y IT A.] — • i •1=1 i T»P / 1=1 T (a) 'Kpot r  E  1  1  \  A  P  where A =  Now  E i-1  A.. 1  introduce  the common f a c t o r i a l f u n c t i o n n o t a t i o n  and l e t <j> = ( p y  (y)  = ^^"^ ,  II A . ) / A . U s i n g the Gauss m u l t i p l i c a t i o n theorem i=l (see, e.g., p. 26 of R a i n v i l l e [I960]) P  P  1  identity  r( a) s P  we  a  r ( a +  ^ ^ \ ^ - \ i  0  F ' }  have  e f(A,a|x,p) = C(x,y)  -A i  i II A. ._, i r A P  x  =  P  P  k (1 + -) •• , , p a-1 — ~ * (D :(ct-l)T a-1 _  1  n  n  ,  P  where C(x,y) depends o n l y on x and fi,,and i s • such", t h a t .  E / f (A,ct |x,y)dA = 1. ot=l Summing w i t h A^'s,  respect  to a g i v e s the j o i n t p o s t e r i o r d e n s i t y of the  g i v e n y and the data  x,  61  -A f ( A | x , y ) = C(x,y)  i  P*  n 1=1  i  P"  A.  k=l  n  a=l a p x. e -A- „ ,Al = C(x,y) _ _ i = i : A j=0  n  r)  (i +  P a-1  2  p-1  n  r).  (l +  k=l  '  (l)P" j! 2  t h a t the i n f i n i t e sum here i s a g e n e r a l i z e d  1 + p-1  1 +  ,a-l <P  (1)P ( a - l ) ! a-1  1  Notice  k  1  hypergeometric  function  1 +  p-2 1 1  1• J- , . . . , _L ,  -L,  p  X  i  which converges f o r j <j> | •> 1, i . e . f o r [ p iy= lE (.~f )\ !• S i n c e the geometric mean o f the A^'s cannot exceed t h e i r a r i t h m e t i c mean, i t i s P  s u f f i c i e n t f o r the f u n c t i o n to converge i f y<l.  L  The j o i n t  d e n s i t y o f X i s proper i f t h e sum o f the o b s e r v a t i o n s i s a r e a s o n a b l e assumption. normalizing  constant  C(x,y).  where  X. = A0.  x  and  •  >• (A, 6 ,...,0 ) 1 p-1 i=l,...,p,  x  E 6. = 1 , 6. > 0. i-1  Jacobian  o f the t r a n s f o r m a t i o n  i sA  /f(A|x,y)dA^dA2 •*dA^ = 1 i m p l i e s i  i s p o s i t i v e , which o f the  We make the f o l l o w i n g t r a n s f o r m a t i o n o f  1  The  posterior  Now we f i n d the e x p l i c i t e x p r e s s i o n  variables: (A , •. ., X ) 1 p  <  p-1 that  62  P-  [C(x,y)]  -x l  z-i  p  - A  = /  A>0  n  i °°  x  / e - A V . n i e . 9> 0 i=l E9.=l l  E j=0  1  ±  k=i  E [p y] P  P  3=0  where z =  9 p  D  2  j !  1  p  [p P y  p  n 9. ] J dA  i=l  1  n de. 1=1 1  P  K  «  3  p  _  ( l )  n (i+ - ) . (z-1)!  ( l+ - ) .  2  P-1 -  k  1  (i)  p  3  2  n  r ( j + x. + l )  1=1 j ! r( j + z +  )  P  P  E x. 1=1 3  T ( j + x. + 1) Now observe that  r( j P  (x. +.„!) . = i ' - " j  + z + p) = r ( z + ) P  and  r ( x . + 1)  (z + p) . = r ( z + PJ  ( c f . p. 22, Lemma 6 of R a i n v i l l e  P  )  p  that  P-1 n (z k=0  j  P  +  p  +  k  p  ).  J  [I960]).  Consequently, we see t h a t P-1  P  (z-1)! n [C(x,y)]~  1  =  xl ±  . , K}. (z+p-1)!  E 2  =  n (1 + -) . n (x:.: + 1) . 31=1 y3  k=l  P  _p^l . (I)?" n (^2%. k=0  0  (z+p-1)!  1 + P 2 p - l 2p-2  i  j  2  p  2  ( z - 1 ) ! n x.!  2  1  1 + ^  2  x +1 1  P  i ' z±P.  Z + 2  x +1; P  P~1 .  P  P  That i s ,  C(x,y) =•  L •  (z+p-1)! 2p-l (z-1)!  n x.! 1=1 1  * • • • »1 Pp" '  pP  •  •  X-H~X  1  »  2p-2 z+p_ 1, . . . , 1, , p  z+2p-l p  •  •  •  » X  P  1  -1  63  For s i m p l i c i t y , we 1 + -jjj- ,...,  use  1 +  a n  the symbol p* to r e p r e s e n t the f i n i t e d  u  se  the symbol 1^ to r e p r e s e n t the  sequence  finite  sequence of q l ' s . With t h i s n o t a t i o n , C(x,y) becomes  p*, x  + l,...,x  x  +  —1  1;  (z+p-1)! 2p-l  2p-2  ( z - 1 ) ! n x.! i=l  l  z+2p-l  p  p  V  1  The  z+p_  -2>  p o s t e r i o r means of the components of A, g i v e n x and  y, a r e  now  e a s i l y shown to be  A  = E  i  x,y  A  =  i  ( »") C(x+e^,y) ' c  l  x  i  =  '  where e. i s a p - v e c t o r which has  i  I  c o o r d i n a t e s zero, and x and  y.fixed.  geometric  E  th  c o o r d i n a t e one  and  the  other  means t h a t the e x p e c t a t i o n i s taken  x, y  holding  T h i s e x p r e s s i o n can a l s o be w r i t t e n i n terms of h y p e r -  functions.  A c c o r d i n g l y , l e t g(x,y) denote the g e n e r a l i z e d  hypergeometric f u n c t i o n , p*, x.+l,...,x 1  g(x,y) :=  2  p  l  ;  1  F ^ 2  P  +1;  2  z+p P-2'  '  P  '  z+2p-l . P '  In terms of g, the p o s t e r i o r mean of A becomes  X. = i  z(x^l) ; z+p  g(x+e.,y) —7 r — , i = l , .. ., p, where z = g(x,y)  The p o s t e r i o r v a r i a n c e s and can a l s o be o b t a i n e d .  p E I  =  1  x. . I  c o v a r i a n c e s among the components of A  In terms of g, they a r e g i v e n as f o l l o w s :  64  Var  (A.) = E A - [E A . ] 1 x,y; 1 x,y 1 2  x,y  2  z(z+l)(x.+l)(x.+2)  g(x+2e ,v)  (z+p) (z+p+1)  g(x,y)  (z+l)(x.+2) i  Cov  x,y  E  ±  '• x,y i"'  g(x+2e.,y)  (z+p+1)  2  E  A  z(x.+l)  g(x+e ,y)  z+p  g(x+e.,y) '  g(x,y)  '  (A.,A.) i 'j  A.A. - (E A . ) ( E A.) x,y:.i j x,y x x,y j  z ( z + l ) ( x ^ l ) (x.,+1)  g(x+e +e ,y)  (z+p>(z+p+l)  g(x,y)  ""  i=l,...,p,  i  „ „  i  "• i j X  X  j=l,...,pi i ^ j  We see t h a t the A.'s a r e c o r r e l a t e d i f a t l e a s t one o f the x.'s i s nonl I zero.  Moreover, s i n c e t h e v a r i a n c e s a r e always non-negative, we have (z+l)(x +2) i  (z+p+1)  g(x+2 ,y) ei  g(x+e y) — i}  z(x +l) ±  ZZZ— ' * z+p  g(x +e ,y) 1  i  „\"'— g(x,y)  i=l,...,p.  That i s , f o r f i x e d i the m a r g i n a l p o s t e r i o r mean E A. i s a nonx,y x d e c r e a s i n g f u n c t i o n of x^ (an i n t u i t i v e l y r e a s o n a b l e r e s u l t ) . . R e c a l l t h a t the f o r e g o i n g  r e s u l t i s based on t h e assumption t h a t a  follows  a t r a n s l a t e d geometric d i s t r i b u t i o n .  S i m i l a r r e s u l t s can be  derived  w i t h d i f f e r e n t d i s c r e t e p r i o r d i s t r i b u t i o n s of a .  Two examples  are g i v e n below. Example 5.2.1. Suppose the p r i o r d i s t r i b u t i o n o f a i s P o i s s o n w i t h c o u n t i n g a-1 d e n s i t y p r o p o r t i o n a l t o exp(-y)y , a=l,2,...,0 j< y < °° and y i s  65  known.  The n o r m a l i z i n g c o n s t a n t i n t h i s case i s  _ C(x,y) =  (z+p-1)! 2p-l 2p-l ( z - 1 ) ! E x. i=l  z+2p-l iPr  1  The  -l  p*, x + l , . . . , x +1; 1 P  p  P  1  p o s t e r i o r mean and the c o v a r i a n c e m a t r i x of X can then be c a l c u l a t e d .  E(A.|x,y)= i 1  z (x.,+1) r ^ — z+p  g(x+e.,y) -.—^ g(x,y)  i=l,...,p,  p*, Xj+1,...,x +1; where g(x,y) = 2 p - l ^ 2 p - l  1 p-1  z+p_ p  z+2p-l p  Example 5.2.2. Suppose the p r i o r d i s t r i b u t i o n o f a i s n e g a t i v e b i n o m i a l w i t h m+a-2'. a-1  counting density p r o p o r t i o n a l to  y" ^(l-y) »a=l,2,..., f o r some m  known m and y such t h a t m >_ 1 and 0 <^ y < 1.  The j o i n t p o s t e r i o r  d e n s i t y o f X g i v e n m,y and x i s  n  e  i=l f(A|m,y,x) = C(x,m,y)  P  n x. i-i  yA  P*. m;  1  1A  where <j> = p  A.i F  P  P P-1  P-1'  -p  1  The n o r m a l i z i n g c o n s t a n t , C(x,m,y), i s g i v e n by P*> x.j+1,... ,x +1, m; C  (x,m,y) •  { ( z - 1 ) ! n.• x.! i - 1 I 1  F 2p 2 p - l  1  LP-1'  _5±P_ P  -1  z+2p-l p  ;  66  The p o s t e r i o r  mean g i v e n x, m and y, i s z(x +l) T •  g(x+e ) 7 ^ — v , i .=• l , . . . , p ,  ±  E( .. x,m,y) = A  1  ' '  i > m ? y  :  z+p  g(x,m,y) p*,  where  •  x.,+1,... ,x +l,m; 1 P  g(x,m,y) = 2p 2 p ~ l z+p_ P-1'  In many s i t u a t i o n s translating  P  z+2p-l. "  it.';is d e s i r a b l e to have g r e a t e r f l e x i b i l i t y i n  one's p r i o r b e l i e f s i n t o a p a r a m e t r i c f a m i l y o f p r i o r  distributions.  A f a m i l y which i s r i c h e r  be s u f f i c i e n t .  Accordingly, richer  the  " '  ?  i n parameters w i l l  results  usually  can be o b t a i n e d by c h o o s i n g  p r i o r d i s t r i b u t i o n o f a to have the f o l l o w i n g  (hypergeometric) form:  . .Vj>k-i k - i = ) a W nV . , ... (k-1) a  P(a  k  k=1>  -Yvk-1  J=l  f o r some known ( a ^ , . . . , a , b^,...,b , y) u  where:  ri  a ^ , . . . » ^  (1)  w =  U  J  F u v  i s well defined, b u  h •  -J^ > • • • 5  U  J  u (2)  n(d.)^  j=l  J  = 1, i f u=0,  (5.2.3)!  67  (3)  u <_ v+1  and  u >_ 0,  if u <  0 <_ y < 1,  v+1,  if u =  L e t a denote ( a , , . . . , a ) and 1 u  v+1.  b denote ( b - , b 1  p o s t e r i o r d e n s i t y of A g i v e n a, b, f(AJa,b,y,x) e  -A  y and  i n A. .1=1 P  X  v  ).  The  resulting ioint J  x is  P*,  a;  1  = C(x,a,b,y)  p+u-1  A  where <}> =  and  p  1  P  b;  -1  p*,  ( z - 1 ) ! n x.! 1=1 (z+p-1)!  x^+1,...>Xp+l, a;  .. 2p+u-l 2p+v-l  1  F  P-1'  In Example 5.2.1, we  had  u=l,  As b e f o r e ,  a-^ = m,  P-1'  y>.A -P  1  [C(x,a,b,y)]  p+v-1  0£y<l.  p o s t e r i o r d e n s i t y , and  u=v=0 and  0<y<°°.  z+p_ z+2p-l p »••*•» p '  to s i m p l i f y the r e p r e s e n t a t i o n  to a i d i n u n d e r s t a n d i n g i t s b e h a v i o r ,  [p*» 2  p  +  u  _ F 1  2  p  +  v  _  '  In Example 5.2.2, we  the g e n e r a l i z e d hypergeometric f u n c t i o n g as  g(x,a,b,y) =  b  we  had for  v=0, the  define  follows:  x^+1,...,x +1,  a; (5.2.4)  1  z+p_ P-1'  P  z+2p-l  68  ;; can [Cov  I n the case o f the hypergeometric f a m i l y o f p r i o r d i s t r i b u t i o n s , we express the p o s t e r i o r mean v e c t o r x,a,b,y  (A., A.)J i n terms o f g. x' J pxp  z(x.+l) ±_— z+p  g(x+e.,a,b,y) — — ± - r — r — , g(x,a,b,y)  z(z+l)(x +l)(x +2)  2  ^  Indeed, '  5  (1) E . A. = x,a,b,y l  E  and the p o s t e r i o r c o v a r i a n c e m a t r i x  i  x,a,b,y i  p,  (5.2.5)  g(x+2 ,a,b,y)  i  ei  (z+p)(z+p+1)  X  i=l  g(x,a,b,y)  (3) Var , (A.) = E , A - [E A. ] x,a,b,y l x,a,b,y l x,a,b,y x 2  '  2  p  (z+l)(x..+2) -  I  E „ i . , . , ~x ,a,b,y"i A  H -  J 1  _  (z+p+1)  g(x+e.,a,b,y)  1  1  z+p+1  g(x,a,b,y)  z(z+l)(x +l)(x +l) ( 4 )  C o v  x,a,b,y  ( A  i  , A  j  )  -[E  ]  '  g(x+e +e ,a,b,y)  i  i  (z+p)(z+p+1)  =  g(x+e ,a,b,y) i  z(x.+l)  1  g(x+2e.,a,b,y) -  1  '  1  g(x,a,b,y)  , A.][E , A.], x,a,b,y I x,a,b,y j  i=l,...,p,  j=l,...,p, i£j.  The m a r g i n a l p o s t e r i o r mean, E , A., i s seen t o be a non-J: 'x,a,b,y l d e c r e a s i n g f u n c t i o n o f x.. x  This follows d i r e c t l y  The MLE o f A^, x^, i s o b v i o u s l y  J  s i n c e Var , (A.) > 0. x,a,b,y i.. —  an i n c r e a s i n g f u n c t i o n o f x..  S i n c e the p o s t e r i o r moments i n v o l v e the f u n c t i o n g ( x , a , b , y ) , we would l i k e t o i n v e s t i g a t e the p r o p e r t i e s o f g. like  to compare  I n p a r t i c u l a r , we would  the v a l u e g(x+e^,a,b,y) and the v a l u e g ( x , a , b , y ) .  s u f f i c i e n t c o n d i t i o n t h a t g(x+e^,a,b,y) i s g r e a t e r  than or e q u a l t o  A  69  g(x,a,b,y) i s t h a t each i n d i v i d u a l term o f the i n f i n i t e  sum o f  g(x+e^,a,b,y) i s g r e a t e r than o r e q u a l to the c o r r e s p o n d i n g term o f the infinite  sum o f g ( x , a , b , y ) .  I n o t h e r words, g(x+e^,a,b,y)  f^g(x,a,b,y)  P p-1 . n (1+J-) • i p a ( 1 )  n (x.+l) • j=l • (x.+2) • ... l a 3  P-1  I  .  (J5+P+JS)  R=l  p-1 •  1  2zl  >;  P  a  .1 k=0  n (b.)  j=l  3  01  u  n (x.+l)  •  P  n (a.)  1  =1  p-1 ^ J E + P + k )  for  .  a  p  n  u n (a.) . i j a  a  .  I  (  j=l  a  = 1  J  h  )  a  a=0, 1,... . The c o n d i t i o n can be s i m p l i f i e d to (x^+2+a-l) z+2p+(a-l):  (x +l) ±  P  -  (z+p) ' ° f  ra  =  1  ' '--' 2  which i s e q u i v a l e n t to * > x. . p - X  p E ( x . / p ) , i s s t r i c t l y g r e a t e r than x., i=l . then g(x+e ,a,b,y) S g ( x , a , b , y ) , and g(x+e ,a,b,y) = g(x,a,b,y) i f P P  Consequently,  i f the grand mean,  1  1  ±  E x./p = x.. j-1  J  i  On the o t h e r hand, i f E j-1  1  g(x+e^,a,b,y) < g ( x , a , b , y ) .  x./p < x., then 3  1  I n f a c t , i f the grand mean i s l e s s than x^,  g(x,a,.b,y). i s a s t r i c t l y d e c r e a s i n g i f u n c t i o n ,of x^. Then Now the msuppose a r g i n a l t ph oa s means t t earli lo rthe _ / saarree i d e n t i c a l and e q u a l t o x^, say. x  70  zCx^l)  g(x+e ,a,b,y)  z+p  g(x,a,b,y)  x,a,b,y I  i  0  T h a t - i s t o say, E A. e q u a l s the MLE i n t h i s c a s e . x, a, b, y l natural result.  .'For, the assumption  This i s a  o f h a v i n g an exchangeable  d i s t r i b u t i o n o f X i m p l i e s t h a t our p r i o r b e l i e f s about they a r e " p r o b a b i l i s t i c a l l y c l o s e " to one a n o t h e r .  prior  t h e _^'s i s t h a t A  Then w i t h  identical  o b s e r v a t i o n s o f t h e x^'s, we have s u p p o r t i n g e v i d e n c e t h a t the ^ ' A  v e r y c l o s e , o r even i d e n t i c a l to one a n o t h e r . XQ  s  a  r  e  Hence, t h e grand mean  i s an a p p r o p r i a t e e s t i m a t e f o r each o f the A^'s.  5.3 M a r g i n a l P o s t e r i o r D e n s i t y In t h i s s e c t i o n , we w i l l d e r i v e the m a r g i n a l p o s t e r i o r d e n s i t y o f  X^, A^'s  f o r 1=1,...,p.  R e c a l l t h a t t h e j o i n t p o s t e r i o r d e n s i t y o f the  i s g i v e n by  -A f(A|x,a,b,y)  = C(x,a,b,y)  1 where p* = (1 H  x  nA,'  i=i  a;  1  p+u-l p+v-1  2 , 1 H  P  = y;p  P  , ,. P  P  n  i=l  ,-P  and .'.A =  E ."A .. i=l 1  71  o f A can be thought o f as a p r o d u c t o f t h r e e « P x. —A i f u n c t i o n s o f A . The f i r s t f u n c t i o n , e n A . , i s maximized when . , x x=l A . = x., i = l , . . . p . The second f u n c t i o n , — ; i n c r e a s e s as the A . ' s ,P i :A d e c r e a s e . . The t h i r d f u n c t i o n . The  1  j o i n t posterior density  1  p*, a; p+u-1 p+v-1  i s maximized when the A^,'s a r e e q u a l to one a n o t h e r .  Hence the p o s t e r i o r  mode i s s h i f t e d from x to a p o s i t i o n where the A_^'s a r e more n e a r l y e q u a l and towards the o r i g i n . The  p o s t e r i o r mean can be used as an e s t i m a t o r  o f A; i t l i e s  between the MLE and the grand mean x-.  To see t h i s , we u t i l i z e  representation  Conditioning  o f the p o s t e r i o r mean.  the p r i o r i n f o r m a t i o n ,  i s therefore  x  x+d  x. + X  x+d  x,  The p o s t e r i o r mean o f A . x r  equal to  x,a,b,y^i  where E i  on a, the d a t a , and  the p o s t e r i o r mean.of A^.'is.  which i s a convex combination o f x. and x. x  another  ^a|x,a,b,y  x. +  x+a  X  i s the e x p e c t a t i o n  d i s t r i b u t i o n o f a.  x+a  x  taken w i t h r e s p e c t  Now, f o r a l l a, the e x p r e s s i o n  l i e s between the MLE x_^, and the grand mean x. of A. a l s o l i e s between x. and x.  to the p o s t e r i o r  i n s i d e the b r a c k e t s  Hence the p o s t e r i o r mean  T h i s does n o t mean t h a t the p o s t e r i o r  mean o f A. an e s t i m a t e o f A. i s n e c e s s a r i l y b e t t e r x x • J  than the MLE x. f o r a x  72  particular i .  We o n l y expect t h a t most of the p o s t e r i o r means o f  X^j i = l , . . . , p , a r e c l o s e r to the t r u e parameters than t h e MLE.  For a  p a r t i c u l a r A^> the squared e r r o r l o s s i n u s i n g the Bayes e s t i m a t o r may be g r e a t e r than t h a t o f the MLE. is  expected  However, t h e t o t a l squared e r r o r  loss  t o be much s m a l l e r than t h a t o f t h e MLE when t h e A / s a r e  " p r o b a b i l i s t i c a l l y c l o s e " t o one a n o t h e r .  On the o t h e r hand, when t h e  X^'s a r e spread a c r o s s a wide range, t h e Bayes e s t i m a t o r i s n o t expected to  p e r f o r m much b e t t e r than the MLE.  the MLE i s b e t t e r .  I n f a c t , t h e r e may be cases where  Such a case i s d i s c u s s e d i n S e c t i o n 7.  Next, we f i n d t h e m a r g i n a l p o s t e r i o r d e n s i t y f o r A_^ i n s e v e r a l instances, f i r s t , in  general.  when the d i s t r i b u t i o n o f a i s degenerate, and then,  R e s u l t s f o r v a r i o u s s p e c i f i c d i s t r i b u t i o n s o f ct w i l l be  studied numerically. Consider f i r s t , i.e.  the case when the d i s t r i b u t i o n o f a i s degenerate,  p(a=s) = l , s > 0.  The p o s t e r i o r d e n s i t y o f A g i v e n x and s i s  e  -A.  ^ [ n \*  f ( A | x , s ) = C(x,s)  P  X  +  s  1  ]r( s) P  — r (s)A P  In  -  P S  o r d e r t o f i n d t h e m a r g i n a l p o s t e r i o r d e n s i t y o f A_^, we i n t r o d u c e the  following notation: D e f i n i t i o n s (1) — ~  (2)  z,.. = (x)  -A,.. = (1)  (3)  e  a )  Ex. i H  3  E A.  iH  3  = ^ , . . . , 6 . ^ ,  9.+1,...,0  )  Now  The  transform  A according to:  (1)  A. +:-A. x x  (2)  A. ->•  j  u;  where  3  0, = 1 and j 1 i .  E  k  Jacobian of t h i s transformation i s A  P  2  (x)  ^^y  f ( A | x , s ) becomes f ( A ^ ,  (^| > )  9  x  s  ;- (i)„ A  X D + ^ P "  A  -A. e  x.+s-l A. l  x.+s-l  P  A f t e r the t r a n s f o r m a t i o n , '  J  n 0.  1  ) -  1  (1)  (A. + x  A,.,) :(x)  P S  (Note t h a t f ( - ) i s b e i n g used g e n e r i c a l l y t o denote the d e n s i t y o f t h e indicated  arguments).  I n t e g r a t i n g out  and (£)> 8  w  e  have the m a r g i n a l p o s t e r i o r  d e n s i t y of A_^ g i v e n by  fUjx.s)  °=  -A. x . + s - l ^"A^ II(X), 1  e  where  Z/ .N + ( p - l ) s - l  -y =  v r  dy.  r  0  1  Let t = ( y / A ) . i  x.  (x)  (X, + y )  Then  p  S  1 1 ( A ) becomes  dt.  ( l )  o  (i+t)  ps  74  Hence the m a r g i n a l p o s t e r i o r d e n s i t y of A. may  1  X: /  \  f(A. • 1  It  1,  s x,s) « e  I  i s reasonable  -At z-1  A.  0  1  e  /  z  ±(p-l)s-l  t (l+t)  dt.  p S  to assume t h a t l a t l e a s t one  hence, z > 0 i s a r e a s o n a b l e  be w r i t t e n  assumption.  of the x.'s J  i s non-zero  and  In t h i s case, f ( A ^ | x , s ) i s a  proper d e n s i t y . Let  us examine the p r o p e r t y of the m a r g i n a l p o s t e r i o r d e n s i t y  f ( A ^ | x , s ) of A^ g i v e n x and If  z^^  I  < s, then the  .(A.) S  "  1  1  = /" 0  s. integral  dt  5  (l+t)  i s f i n i t e even when A. i s z e r o . l x.  i  f(A.|x,s) - e  -A. X  1  and to A.. x  A. ~i  -X  X  f°°oo -Q  o„  (s > 0)  p s  I f s = l = z . , we (x)' /  . p-1  -A.t X  ^+-  1  (i+t); „..vP  x  have  dt ,  p  '  f(A_Jx,.s) i s s t r i c t l y d e c r e a s i n g i f x  = 0.  I t i s straightforward  check t h a t the i n t e g r a l i s a monotone d e c r e a s i n g , convex f u n c t i o n of Moreover, f ( A . | x , s ) tends x 1  to i n f i n i t y  i s not s u r p r i s i n g because x_^ = 0 and very l i k e l y  to be near z e r o .  as A. tends x  In g e n e r a l , we would expect  the curve of f(;A |x,s) i s a g a i n unimodal and  suggested  we  the shape of  I f x^ > 0 and  skewed to the  c o n s i d e r the cases i n which a has  i n t h i s paper.  T h i s .'.  z = 1 would i n d i c a t e ..that A_^ i s  the m a r g i n a l p o s t e r i o r d e n s i t y to be gamma-like.  Now,  to z e r o .  the d i s c r e t e  s ^  1,  right. distributions  The d e r i v a t i o n of the m a r g i n a l p o s t e r i o r  75  d e n s i t y of A ^ i s s t r a i g h t f o r w a r d but tedious.  f(A  |x,a,b,u)  P-1 -A. i , z-1 e :. A . . E i "-  p  n (x.+l)  p S  ,  n  3  =  0  u  s  t  r  S  n  n  ) P-1  k i = 0  • I (A.)  s  J  N  (D^si s  ,  •:. Tn. (a.) -IT (1+-;) p s i s :•: y k=l i'= l "• " " " p-2: z ,. +p-l+k / x v  . 1 . 3  .1^1  1  a  In f a c t ,  • (p-i)  -  ( p  1 ) s  • n(b.)  s  3 s  D e f i n e the hypergeometric f u n c t i o n h ^ ^ ( x , y , t ) p  p-1  i  n (x.+l) . )s •j.3 s • ,n T ( i +p-s) .:• . n .( i Jfi k=l j=l P-2 z,..+p-l+k v CD* - 1 ) ,. n (b.) s k=o P" j=i J  =  E  y  n  3 = 0  p  a  J  n (-^4 1  s  J  < -i) - (i+t) (p  1)  p  P  s  p*, x.,+1,... . x ^ + l , x  i + 1  + l , . . . ,x^+l,a; w  2p-2+u 2p-2+v z,. +2p-3 (i) p-1 x  L  where w =  p-1'  p-1  , b;  ^P-—  (p-D^-^d+t)  15  We then have f o r t h e g e n e r a l (non-degenerate  a) m a r g i n a l p o s t e r i o r  density, , i f ( A . |x,a,b,y) <= e 1  -A i  z-1 oo A. / 0  e  1  X  t  z...+p-2 t  (  l  : (l+t)  )  • Yi.(x,y, P  W  t)dt .  I  76  I t can be. checked t h a t the n o r m a l i z i n g c o n s t a n t i s  c(x,a,b,v);  n r(x.+i)/r(z,..+p-i)  -1  p*, x.j+1,... ,x +1, a; (z+p-1).! P  2p+u-l 2p+v-l 1  ( z - 1 ) ! n x.! j=l  3  ^±P_ z+2 -i P ' "*' P P  b  .  n x.!/(z,..+p-2)!  (z+p-1)! (z-l)!(z +p-2)!x.! ( i )  -p*, x.+l,...,x +1, a; 1 p  -1  2p+u-l 2p+v-l z+p_ p-1'  z+2p-l  P " " '  P  The shape o f t h i s d e n s i t y f u n c t i o n i s a l s o gamma-like, by the same argument as b e f o r e .  5.4 Summary In t h i s s e c t i o n we proposed Bayes e s t i m a t o r s  f o r simultaneously  e s t i m a t i n g the parameters of p d i s t r i b u t i o n s of independent P o i s s o n  ran-  dom v a r i a b l e s when the p r i o r d i s t r i b u t i o n o f the parameters i s assumed to be exchangeable.  S u b s t a n t i a l improvement over the u s u a l procedure  (which i s the MLE) i s expected when the parameters a r e c l o s e t o one another, e s p e c i a l l y when p i s l a r g e , because t h e assumption of e x c h a n g e a b i l i t y i m p l i e s t h a t the l a r g e r p i s , the more i n f o r m a t i o n we have about the A ^ ' s .  Our c l a i m i s supported i n S e c t i o n 7 by t h e r e s u l t s  77  of a computer s i m u l a t i o n designed  to compare the e s t i m a t i o n e f f i c i e n c y  of the Bayes e s t i m a t o r s w i t h t h e MLE.  The measure used to a s s e s s the  performance of the e s t i m a t o r s i s mean squared  e r r o r , which i s o f t e n a  r e a s o n a b l e measure o f the o v e r a l l adequacy of an e s t i m a t o r .  78  SECTION 6.  6.1  EMPIRICAL BAYES ESTIMATION  Background In e s t i m a t i n g the mean of a m u l t i v a r i a t e normal random v e c t o r ,  E f r o n and M o r r i s [1973] g i v e an i n t e r p r e t a t i o n of the e s t i m a t o r from an e m p i r i c a l Bayes p o i n t of view. p  V.~  •(V.,...,V ) ^ N ( 6 , I ) , S 1 P P P  =  E i  =  1  - (p-2)/S)V.  =  i d e n t i c a l l y ^ N(0,A).  estimator of 9 i s  In t h i s s i t u a t i o n ,  (1 - B)V, w i t h B = 1/(1 + A ) .  d i s t r i b u t i o n of V i s N ( 0 , ( 1 + 4 ) I ) and p  p  The  B by B(S) = (p-2)/S, which i s an u n b i a s e d  the  marginal  S i s d i s t r i b u t e d as  m u l t i p l e oJ a"chi-square d i s t r i b u t i o n w i t h p degrees  (l+A) \ a X p  o f freedom.  Replacing  e s t i m a t e of B, y i e l d s  e m p i r i c a l Bayes e s t i m a t o r 9 = (1 - B ( S ) ) V which i s the estimator.  i s the  E f r o n and M o r r i s assume t h a t the c o o r d i n a t e s 9^ o f 9  a r e i n d e p e n d e n t l y and Bayes r u l e i s e*  I f we l e t  2 V. and p > 3, where I i P  p x p i d e n t i t y m a t r i x , then the James-Stein XI  James-Stein  the  James-Stein  E f r o n and M o r r i s p e r f o r m much of t h e i r a n a l y s i s based  what they d e f i n e as the " r e l a t i v e s a v i n g s l o s s " , i . e . the r e g r e t  on from  u s i n g the e m p i r i c a l Bayes r u l e 9 i n s t e a d o f the a c t u a l Bayes r u l e d i v i d e d by the c o r r e s p o n d i n g r e g r e t i f the MLE, the Bayes r u l e .  V, i s used  i n s t e a d of  I f t h e r i s k o f the e s t i m a t o r 9 i s denoted  by  ..  R(B,9) = E,, R(9,9), where E.„ i n d i c a t e s t h a t e x p e c t a t i o n i s taken w i t h r e s p e c t to the above p r i o r d i s t r i b u t i o n , RSL(B,9) =  the r e l a t i v e s a v i n g s l o s s i s  [R(B;9) - R(B,9*)]/[R(B,V) - R ( B , 9 * ) ] .  Under the above assumptions,  s t r a i g h t f o r w a r d c a l c u l a t i o n s show t h a t  RSL(B,9) can be w r i t t e n as RSL(B,9) = EL' {(B(S) - B) / B } . 2  Here  a  D  2 i n d i c a t e s e x p e c t a t i o n w i t h r e s p e c t to S ^  (1/B)x  + 9  .  The  empirical  -  79  Bayes approach thus reduces the p-dimetisional problem o f e s t i m a t i n g 0 = (0.., ...,0 ) from V = (V.,... ,V ) t o the one-dimensional problem o f 1 p 1 p e s t i m a t i n g B, o r more p r e c i s e l y ,  6.2  1/(1+A).  R e l a t i v e Savings Loss i n t h e P o i s s o n Case In  t h i s s e c t i o n , we use the n o r m a l i z e d squared e r r o r P (A. - A . ) L(A,A) = Z 1=1 i 1  loss  2  1  as our measure f o r the l o s s i n e s t i m a t i n g A^ by A l , i = l , . . . , p . i s the l o s s f u n c t i o n employed by Clevenson and Z i d e k [1975].  This  As they  suggested, the above approach used on the m u l t i v a r i a t e normal e s t i m a t i o n problem seems a p p l i c a b l e t o our problem o f s i m u l t a n e o u s l y e s t i m a t i n g the parameters  A^,...,Ap o f t h e independent P o i s s o n v a r i a b l e s X^,...,X^.  Our developments  here p a r a l l e l those i n the normal  case.  A l t h o u g h no new  e s t i m a t o r s a r e found, t h e e m p i r i c a l Bayes approach p r o v i d e s an a l t e r n a t e way t o view our problem and d e r i v e the e s t i m a t o r s g i v e n i n Theorem 2.2.9 so t h a t we have a b e t t e r u n d e r s t a n d i n g o f those e s t i m a t o r s . We w i l l next d e r i v e the " r e l a t i v e s a v i n g s l o s s " f o r the P o i s s o n case.  L e t A^,...,A  be i n d e p e n d e n t l y and i d e n t i c a l l y d i s t r i b u t e d w i t h  2 prior distribution a ^ j  scalar multiple of a chi-square d i s t r i b u t i o n  a  w i t h two degrees of freedom. The j o i n t p r i o r d e n s i t y o f A.,...,A i s P P p r o p o r t i o n a l t o e x p ( - Z A./a) dA,•••••dA. f o r A. > 0, and zero o t h e r w i s e .  1-1  1  1  ?  The Bayes e s t i m a t o r o f A when such a p r i o r i s used i s ( l - b ) X , where b = 1/(1 + a ) .  T h i s e s t i m a t o r has Bayes r i s k p ( . l - - b) , w h i l e the MLE,  X, has r i s k p.  Thus, the r e g r e t a t u s i n g t h e MLE i n s t e a d o f the Bayes  e s t i m a t o r i s pb.  The m a r g i n a l d i s t r i b u t i o n of Z =  P Z X., g i v e n b, i s i=l 1  80  the n e g a t i v e b i n o m i a l d i s t r i b u t i o n NB(b,p) w i t h p r o b a b i l i t y mass f u n c t i o n Pr(z|b)  p z + b.(1-b) , f o r z e J .  z+p-1 z  =  estimated by b ( Z ) .  Then the r e g r e t a t u s i n g the e m p i r i c a l Bayes •:.  e s t i m a t o r X = ( l - b ( Z ) ) X i n s t e a d of the Bayes r u l e  E  b  E  x  [  Z ([1 - b ( Z ) ] X . - X . T i=l  = E ' (1/(1-6))  [Zb  b  =  Suppose b i s to be  / X,]  (1 - b)X i s  - p ( l - b)  (Z) - 2b(Z)Z + 2 b ( Z ) Z ( l - b ) ] + p - p ( l - b )  [{ Z(b(Z) - b )  Z  } / (1 - b)]  where E^" denotes e x p e c t a t i o n w i t h r e s p e c t to the m a r g i n a l of Z as g i v e n above.  The r e l a t i v e s a v i n g s l o s s i n t h i s case i s  RSL(b,X) = E ' [ Z(b(Z) - b ) b  =  Z z=0  =  Z z=l  =  z z=0  ^  distribution  (  1(1  ( Z )  -  7 ? b  b)pb  2  / p b ( l - b)]  • z •  (b(z) - b)' (l-b)pb  ( b ( 2 + 1 )  - b>  2  z!  (p-1)!  (z+p-1) ! (z-1)!(p-1)!  .  l & n .  z! p!  b  P  + 1  • b * ((ll-- b ) '  ,p  n  (i-b)  [(b(Z+l) - b) / b ]'  where E^ denotes e x p e c t a t i o n w i t h r e s p e c t to the d i s t r i b u t i o n of Z and Z ^ NB(b,p+l).  81  6.3  The P l u s  Rules have an e s t i m a t o r of the form X =  Suppose we  t h e r e i s a p o s i t i v e p r o b a b i l i t y t h a t b(Z) > 1. t i o n , shows  t h a t we  (1  -  b ( Z ) ) X and  that  The f o l l o w i n g p r o p o s i -  can always improve the r e l a t i v e s a v i n g s l o s s RSL A  A.  j  of  ^  such an e s t i m a t o r by r e p l a c i n g b(Z) w i t h b (Z) = Min { l , b ( Z ) } . I n t u i t i v e l y , we would expect  t h a t the e s t i m a t o r X = (1 - b ( Z ) ) X can  improved upon by r e p l a c i n g b(Z) w i t h 1 when b(Z)  i s g r e a t e r than  because the Bayes e s t i m a t o r i s (1 - b)X, where b  i s known to be  between 0 and 1.  be  1,  Hence the p r o p o s i t i o n i s a v e r y n a t u r a l r e s u l t .  P r o p o s i t i o n 6.3'..l.  X+  Let (1/b ) \  {[  2  The  =  (1 - b ( Z ) ) X .  Then RSL(b,A) - RSL(b,X ) =  +  +  (b(Z+l) - b ( Z + l ) ) + +  f u n c t i o n of Z i n the outermost  g r e a t e r than 0 i f b(Z) strict  (l-b)]  2  -  [1 -  b] }. 2  brackets i s nonnegative  and  strictly  > 1, so t h a t RSL(b,X) >_RSL(b,X ) f o r a l l b, w i t h  i n e q u a l i t y i f Pr {b(Z)  > 1} i s p o s i t i v e f o r any v a l u e of b.  The p r o o f of the above p r o p o s i t i o n i s immediate and hence i s j  omitted.  6.4  The e s t i m a t o r X  /\  may  be c a l l e d  Bayes Rules w i t h Respect In  t h i s s u b s e c t i o n , we  to Other  the " p l u s r u l e " of  X.  Priors  s h a l l c o n s i d e r some Bayes r u l e s w i t h r e s p e c t  to p r i o r s which a r e members of the f a m i l y c o n s i d e r e d by Clevenson Zidek  [1975].  We  reparametrize  the  parameters X^,...,X  as  and  (0 ,A),  P P i = l , . . . , p , where A = E X. and 0. = X./A, and suppose the j o i n t i=l d i s t r i b u t i o n of (0^,A), i = l , . . . , p , i s p r o p o r t i o n a l to 1  1  exp(-A/a)A dAd0 -'W0  (6.4.1)'  a  1  when A >0  and  P E i=l  1  0. = 1, and 1  1  P  zero  otherwise,  prior  82  where a i s a p o s i t i v e i n t e g e r . prior distribution i s The m a r g i n a l  Bayes e s t i m a t o r w i t h r e s p e c t to  = (1-b) (Z+a)X / (Z+p-1), where b =  {Z/(Z+a)} =  the e m p i r i c a l Bayes  this  l/(l+a).  p r o b a b i l i t y mass f u n c t i o n N B ( b , a + l ) .  d i s t r i b u t i o n of Z has  Since &  A*(X)  The  (1-b), e s t i m a t i n g  (1-b)  by Z/(Z+a) l e a d s  to  estimator  = [Z/(Z+p-1)]X = [1 -  which i s independent of a.  (p-1)/(Z+p-1)]X,  T h i s e s t i m a t o r belongs to the c l a s s of  e s t i m a t o r s mentioned i n Theorem 2. 2.9.  The r i s k of \* can be c a l c u l a t e d as  a  follows.  p R(b, A ) = E, E a b A,-0  = E  E  b  Z . ..  Z +  1=1  Cl/A)  z  [ ( l - b W g ^ X. - A 9 , ] P"  1  1  1  f l Q  A9.  2  —  x  •  {(1-b)  2  [Z^l]  2  •  ~ tL-b) ' ^ = 1 2  ZA+A }  where E^ i n d i c a t e s e x p e c t a t i o n w i t h r e s p e c t to the p r i o r s t a t e d above and  Eg i n d i c a t e s e x p e c t a t i o n w i t h r e s p e c t to P o i s s o n E^,^ A = L jD E  z,b  where E  (  1  /  A  [(Z+a+l)/(l+  )  =  1  1  [  b  z + a  i s the e x p e c t a t i o n  L* ) D  d i s t r i b u t i o n of .A  g i v e n Z and  >  J>  taken w i t h r e s p e c t to the p o s t e r i o r b.  Hence  R(b, A ) = E { (1-b) (Z+a+1) - (1-b) (Z+a)a/(Z+pr-1) } a b = p(l-b) +  Note t h a t  ( 1 / a ) ) ] = (1-b)(Z+o+1) and  ( - )( 1  (A).  (l-b)(a+l-p)  [ (p-1)/(Z+p-1) ]  2  83  where  denotes e x p e c t a t i o n w i t h r e s p e c t t o t h e p r o b a b i l i t y mass  f u n c t i o n NB(b,a+l).  F o r example, when a = p,  R(b,A ) = p(.l-b) •+ ( l - b ) [ b  2  p  + (p-1) b (1-b)/p]  = p - b[p - 1 + l / p . + 'Cl - 2/p)b + b / p ] 2  _< p f o r a l l b e ( 0 , 1 ] .  6.5  Truncated  Bayes  Rules  I n t h i s s u b s e c t i o n , we c o n s i d e r a g a i n the c l a s s o f e s t i m a t o r s o f the form (1 - b ( Z ) ) X . dominating  Theorem 2.2.9 p r o v i d e s a c l a s s o f such e s t i m a t o r s  the MLE.  We w i l l attempt to f i n d c o n d i t i o n s under which they  are a l s o Bayes w i t h r e s p e c t to some p r i o r d i s t r i b u t i o n . ^ Poisson  ( A ) independently,  i = l,...,p.  A ^ ^ (1/a) exp ( - A ^ / a ) , and l e t b = 1/(1+a).  As b e f o r e ,  Suppose each Furthermore, we suppose  t h a t b ^ h ( b ) , where h(b) i s a p r i o r p r o b a b i l i t y d e n s i t y f u n c t i o n o f b p u t t i n g a l l o f i t s p r o b a b i l i t y on the i n t e r v a l the Bayes r u l e under n o r m a l i z e d .A*(x)  = E  h  squared  (0,1].  In t h i s  setting,  l o s s i s c a l c u l a t e d to be  [E(A|x,b)]  = /J(l-b)Xh (b) z  db  = [1 - b * ( Z ) ] X  (6.5.1)  where E ^ i n d i c a t e s e x p e c t a t i o n w i t h r e s p e c t t o the s i t u a t i o n d e s c r i b e d above, h^Cb) i s the c o n d i t i o n a l d e n s i t y o f b g i v e n Z, and bg(Z) i s t h e c o n d i t i o n a l e x p e c t a t i o n o f b g i v e n Z.  The Bayes r u l e A* i s thus o f t h e  form t h a t we have been c o n s i d e r i n g and Theorem 2.2.9 i s a p p l i c a b l e .  84  The  f o l l o w i n g lemma g i v e s e q u i v a l e n t d e f i n i t i o n s o f the Bayes r u l e  w i t h r e s p e c t t o h.  I n p a r t i c u l a r , an a l t e r n a t e d e f i n i t i o n which proves  to be more convenient  f o r our a n a l y s i s than t h e u s u a l one i s suggested.  Lemma 6.5.1. The  following are equivalent:  (i)  (ii)  A* i s t h a t A which minimizes J(jR(b,A) h(b) db.  A* i s t h a t A which minimizes /o[R(b,A) - p ( l - b ) ] / p b g(b) db where g(b) = pbh(b).  (iii)  A* i s t h a t A = (1 - b ( Z ) ) X f o r which b ( Z ) minimizes h a. E g  where E  2  b(Z+l) - b  ,  b(Z-El) - b b  2  ' g(b) db  (6.5.2)  means e x p e c t a t i o n w i t h r e s p e c t to Z ^ NB(b,p+l)  Proof: Form ( i ) i s the u s u a l d e f i n i t i o n ; form ( i i ) f o l l o w s d i r e c t l y because the m i n i m i z a t i o n (6.2.1).  i s over A.  Form ( i i i )  i s e q u i v a l e n t t o form ( i i ) by  Q.E.D.  F i n d i n g the Bayes r u l e s having  t h e form A = (1 - b ( Z ) ) X  e q u i v a l e n t to f i n d i n g b which m i n i m i z e s (6.5.2). minimization  in (iii)  b*(z+l) h  j J g ( b ) b ( l - b ) db = 4/ ^ g ( b ) b - ( l - b ) db  (6.5.3)  j ; o h ( b ) b ( i - b ) db = — / o h ( b ) b ( l - b ) db  (6.5.4)  Z  p  1  Z  p + 1  or  b*(z+l) n  The b ( z ) t h a t g i v e s the  i s b * ( z ) , which i s g i v e n by P  P  z  Z  i s thus  .  85  I f we take the improper r u l e i s (1 - b * ( Z ) ) X  p r i o r h(b) = 1/b  2  o r g(b) = p/b, then the Bayes  with  /Jb- b ( l - b ) db b*(z+l)=-j r : / J b - b ( l - b ) db h  2  P + 1  2  P  Z  Z  = (p-1)/(z+p)  or  b*(z) =  (p-1)/(z+p-1).  Thus the e m p i r i c a l Bayes e s t i m a t o r X* -  ( 1 — (p-1)/(Z+p-1))X  derived i n  S e c t i o n 6.4 i s a c t u a l l y a Bayes r u l e . We next i n q u i r e i f t h e Bayes r u l e s thus o b t a i n e d w i l l s t i l l dominate the MLE.  Let $  s  be the c l a s s o f e s t i m a t o r s  {1 - [ (p-1) cp(Z)/(Z+p-1) ]}X d e s c r i b e d i n Theorem 2.2.9 s a t i s f y i n g l i m <p(z) = s <_ 2. The f o l l o w i n g z-x» theorem g i v e s the c l a s s o f e s t i m a t o r s o b t a i n e d by m o d i f y i n g the Bayes r u l e o b t a i n e d i n t h i s s e c t i o n so t h a t they a r e i n $  s  f o r some 0 < s < 2. — —  Theorem 6.5.5. Suppose t h a t h(b) i s such t h a t <P*(z) = b * ( z ) / [ ( p - 1 ) / ( z + p - 1 ) ] i s nonnegative  and nondecreasing  i n z.  The e s t i m a t o r i n $  i m i z e s the Bayes r i s k v e r s u s h, E ^ R(b,A), i s g i v e n by  A*(X) = {1 s where <i> ( ) z  n  =  [(p-1)/(Z+p-1) ] ^ ( Z ) } X  Min {s,(p*(.z) }.  g  which min-  86  Proof: C o n d i t i o n on Z=z, and l e t g (b)=pbh (b) . Then g (b) °= g C b ) b z z z where g(b) = pbh(b) as b e f o r e . F o r any e s t i m a t o r A = (1 - b ( Z ) ) X ,  /J[(b(z+l)-b)/b] g (b)  P + 1  (l-b)  db  2  z  =/o[(bCz±l)-b*(.z+l))/b] g Cb) db + [Cb*Cz+l)-b)/b] g (b) 2  db.  2  z  z  The c r o s s term i s zero because of formula (6.5".3) f o r b * ( z + l ) . h of  I n terms  <p(z) = b ( z ) / [ (p-1)/(.z+p-1)],  the above e q u a l i t y :can be w r i t t e n as  /J'[(b(z+1) - b)/b] g Cb) db 2  z  = jl [(<p(z+l) - <f>*(z+l))]  2  [(p-l)/Cz+p-l)] g Cb) db 2  2  + j] [ ( b * ( z + l ) - b ) / b ] g ( b ) db.  (6.5.6)  2  z  g  I t i s seen t h a t  '('^(z) minimizes  the r i g h t hand s i d e of the above e q u a l i t y  f o r a l l z among cp(z) g i v i n g r u l e s i n $ . g  I n t e g r a t i n g over  the m a r g i n a l  minimizes the i n t e g r a l j  1 0  ~  s  d i s t r i b u t i o n o f Z shows t h a t A. h  RSL(b,A) g(b) db f o r A i n $ . s Q.E.D.  The e s t i m a t o r  thus o b t a i n e d  i s c a l l e d a " t r u n c a t e d Bayes r u l e " , a  term i n t r o d u c e d by E f r o n and M o r r i s [1973]. s s We now d e f i n e b^Cz) = [ (p^l)/(z+p-1) ]cp (z) w i t h s < l i m <j>*(z), and l e t ' A ? ( X ) = (1 - b ? ( Z ) ) X . — z-*-°° n n h b  The f o l l o w i n g lemma  shows t h a t the r i s k of A, i s a d e c r e a s i n g convex f u n c t i o n o f s.  Z  87  Lemma 6.5.7, R(b,A^) i s a s t r i c t l y d e c r e a s i n g convex f u n c t i o n o f s.  Proof: s 2 2 S i n c e {d>, (z) - <p*(z)} = [Max {0,<p*(z)-s}] i s a d e c r e a s i n g n n n f u n c t i o n o f s, the r e s u l t f o l l o w s from  convex  (6.5.6). Q.E.D.  6.6  The R i s k F u n c t i o n of t h e E s t i m a t o r A* As remarked i n t h e p r e v i o u s s u b s e c t i o n , A* = [ 1 - (p-1)/(Z+p-1)]X  can be viewed as an e m p i r i c a l Bayes e s t i m a t o r under the assumption t h a t the p r i o r d i s t r i b u t i o n o f A i s g i v e n by (6.4.1). a , t h i s estimator  S i n c e A* i s independent o f  i s r o b u s t i n t h a t any member of a whole f a m i l y o f p r i o r  d i s t r i b u t i o n s may be chosen and s t i l l we a r r i v e a t the same e m p i r i c a l Bayes e s t i m a t o r .  We t h e r e f o r e proceed  R(A,A*) o f the e s t i m a t o r  A*.  to c a l c u l a t e the r i s k f u n c t i o n  We f i r s t d e r i v e an e x p r e s s i o n f o r t h e r i s k  f u n c t i o n as a f u n c t i o n o f b. As i n s u b s e c t i o n 6.2, we assume t h a t A^,...,A  are independently 2  and  i d e n t i c a l l y distributed with p r i o r d i s t r i b u t i o n a x ' 2  Equivalently,  the p r i o r i s o f the form g i v e n above i n (6.4.1), w i t h a = p - 1. RSL  can then be c a l c u l a t e d as f o l l o w s . RSL(b,A*|a=p-l) = E  [b (Z+l)/b 2  fe  2  - 2b(Z+l)/b + 1]  OO  =  I [(p-1)/(z+p)] z=0  2  [(z+p)!/(b z!p!)]b 2  P + 1  (l-b)  Z  OO  -  E z=0  [2(p-l)/{b(z+p)}][(z+p)!/(z!p!)]b  P + 1  (l-b) + 1 Z  OO  =  E [(p-1) /(z+p)][(z+p-1)!(z!p!)]b z=0 2  P _ 1  (l-b)  Z  - 2(p-l)/p + 1  The  88  CO  = (p-l) b 2  p - 1  E z=0  [l/(z+p)][(z+p-1)!(z!p!)](l-b)  [(p-l) b /{p(l-b) }] 2  p-1  E  p  - 2(p-l)/p + 1  Z  [(z+p-1)!/ z ! ( p - 1 ) ! } ] / J ( l - i )  z=0  Z+P  1  dt  H  - 2(p-l)/p + 1  = [(p-l)V" /{p(l-b) }]/J 1  E [ (z+p-1) !/{z! (p-1) !}] (1-1) z=0  P  0  Z  +  P  _  dt  1  - 2(p-l)/p + 1  = [(p-l) b 2  P _ 1  /{p(l-b) }]/J(l-t) " /t P  P  1  P  d t - 2 ( p - l ) / p + 1.  Observe t h a t as b tends to zero, RSL(b,A*|a=p-l) tends to (p-l)/p - 2(p-l)/p + 1 = 1 -  ( p - l ) / p = 1/p.  RSL(b,b|a=p-l) tends to ( p - l ) / p 2  As b tends to one,  2(p-l)/p + 1 =  2  1/p . 2  U s i n g the above r e s u l t t o c a l c u l a t e the r i s k o f the e s t i m a t o r  as a  f u n c t i o n of b, we have  R(b,X*|a=p-l) = RSL(b,A*)pb +  =  p(l-b)  (p-l) b /(l-b) i7j 2  P  P  (l-t)  P _ 1  /t  P  dt - 2(p-l)b  + pb +  = p - 2 ( p - l ) b + (p-1) V / ( 1 - b )  R e c a l l that t h i s expression not a simple e x p r e s s i o n ,  depends on the c h o i c e  i n the next s u b s e c t i o n  of s i m p l i f y i n g the e x p r e s s i o n  by v a r y i n g  a.  P  p(l-b)  (l-t)  of a.  P _ 1  /t  P  dt.  Since t h i s i s  we e x p l o r e  the p o s s i b i l i t y  89  6.7  S i m p l i f i c a t i o n of R(b,A*) In our attempt  to o b t a i n a s i m p l e r e x p r e s s i o n f o r R(b,A*), we  adopt another p r i o r d i s t r i b u t i o n .  Suppose the j o i n t p r i o r  i s as g i v e n i n (6.4.1), w i t h a = p. prior  now  distribution  The r i s k R(b,A*|a=p) when such a  i s used i s  E  a Ve  = E  *  [((i-Mz))x. - Ae^/CAe.)]  '  i=l  E  ( l / A ) [ Z ( Z + p - l ) { b ( Z ) - b(Z) + 2b(Z)A/(Z+p-1)}] + p  cL  2  Li  ( c o n d i t i o n e d on Z, the ^ ' s have a m u l t i n o m i a l d i s t r i b u t i o n w i t h p a r a x  meters 9_^, i = l , . . . , p )  Z ( Z +  " b  l=b  E  P- T ]  ) B  ^2,  - 2MZ) (Z+P-1)  ( Z )  Z+p  +  2  b  (  z  )  z  (  1  _  b  + P  )  Z+p  where E, i n d i c a t e s e x p e c t a t i o n w i t h r e s p e c t to the m a r g i n a l b of Z ^ NB(b,p+l).  M U  F )  E  With b(Z) = (p-1)/(Z+p-1),  _ I _ b 1-b  1-b  2  z(z+p-i) z+p  n  z=0  I  2  1-b  =  =  i  ^  2 l l _ 1-b  ?  have  Z(Z+p-l)b (Z) Z+p  (P-D  =  we  =  =  Cz+p-2)! p + l  b  (p-l) b/p. 2  T  b  °° lz+E=l)l P+l z!p! Q  (z+p)! ,p+i ;• z!p!  2  (z-l)!p!  E z  (p-i) , v - N2 (z+p-1)  ,z  (  U  ( 1  1  b J  _  b ) Z  +l  ,z  distribution  (ii,  ^  ^  M  -2 1-b  Z  )  ^  „ 2=l_ i£^ P+l . z+p-1 z!p! z=0 z  b  (  1 v  .  b  )  «5!2=l ' z+p  = -2(p-1)  (iii)  2E b(Z)Z b 1  OO  - 2(p-1)  = 2(p-l)  2  (  p  1  )  P  I  bP+Vb) 2  (z+p) < ^ P - > j 2  z (z+p-1) ; ( P - l > zip z=0 z +  !  b ^ d - b ) ^  1  b ( l - b ) [ p ( l - b ) / b + p + 1]  2(p-1) - 2 b ( p - l ) / p - 2 b ( p - 1 ) / p . 2  2  Consequently, R(b,A*|a=p) becomes  p  b ( p - l ) / p - 2 b ( p - l ) / p , 0 « b <_ 1. 2  (6.7.1)  2  E q u i v a l e n t l y , R(a,A*) = p - (p-1) /{p(1+a)} 2  - 2(p-l)/{p(l+a) }, 2  a ^  (6.7.2) A.  A.  R(b,A*|a=p) i s c l e a r l y concave, and d e c r e a s e s i n b from R(0,A*|a=p) to R(l,A*|a=p)  = 1/p.  91  6.8  R i s k F u n c t i o n of A* as a F u n c t i o n of A We  have now  o b t a i n e d a r e l a t i v e l y simple e x p r e s s i o n f o r R(b,A*),  and  P note t h a t R(A,A*) depends on A o n l y through the v a l u e of A = £ A.. . i=l The r i s k f u n c t i o n R(A,A*) i s t h e r e f o r e a c t u a l l y a f u n c t i o n of A, 1  R(A,A*), f o r which we  a r e now  i n a p o s i t i o n to d e r i v e an  expression.  U s u a l l y , knowledge about R(A,A) i m p l i e s knowledge of R(b,A) when the p r i o r d i s t r i b u t i o n of A i s known. r e v e r s e the d i r e c t i o n and  The  r e s u l t below shows t h a t we  can  g a i n i n f o r m a t i o n about R(A,A*) from knowledge  about R(b,A*). With (6.4.1) as the p r i o r d i s t r i b u t i o n  P  (A) =  a  (A/a) [exp(-A/a)]/[ar(p+l)]  (a=p), A has d e n s i t y f u n c t i o n  if A > 0  P  = 0  i f A < 0.  By d e f i n i t i o n , R ( a , A * ) = / Q R(A,A*) P  (A) dA, which can be w r i t t e n as cl  R(a,A*) = /o R ( a A , A * ) A D i f f e r e n t i a t i n g both s i d e s of d R(a,A*)  P  exp(-A)/r'(p+l)  da-  r(p+l)  a=0  However, from (6.7.2),  dA  (6.8.2)  j  A=0  the d e r i v a t i v e on the l e f t  s i d e of (6.8.2) i s  - ( p - l ) ( - l ) j ! / p - 2 ( p - l ) ( - l ) ( j + l ) ! / p , j = !,...,» . 2  T  h  e  r  e  f  o  r  e  J  j  d^R(A A*)  =  y  dA"  A=0  (6.8.1)  (6.8.1) j times w i t h r e s p e c t to a g i v e s  r ( p + j + l ) d R(A,A*)  j  dA.  r( +i) r(p+j+i) P  •2(p-1)  92  R e p r e s e n t i n g R(A,A*) as a power s e r i e s expansion about A = 0 g i v e s  J*) -  R ( A  p  •  I  ^Szll  -  j=o  P  P  (J+Dr(p+D R  (  P+J+  1  (  _A)j  m  (p-D2 * j=o  P  )  r(P+i) r  j  (p J+D +  C  ;  (6.8.3) CO  Furthermore,  ^  Z  ^  ^P^|  1  (-A)  )  (see Abramowitz and Stegun  = /: p exp(-At) ( l - t ) P " d t  j  1  0  [1964], p. 505).  r(  00  The s e r i e s  Z ., (-A) i s u n i f o r m l y convergent on bounded j=0 i n t e r v a l s o f the r e a l l i n e . Hence J  K  -(j+i)r( +i) ' r(p+j+i)  z ^  v 3  )  j  P  0  (  A  )  r(p+D  E  jlo  r( P +j+i)  i f  00  = <L dA  = ^  = p/  z  Ap/J  0  r(P+i) F(p+j+l)  ( c  _ _ (  A )  c A ; )  j+ix >  exp(-At) ( l - t ) P "  exp(-At) ( l - t )  P  1  dt  d t - p/o  The l a s t term can be r e w r i t t e n as -pJ  .1 0  t exp(-Af)(l-t)  exp(-At) t ( l - t )  = - p [ - e x p ( - t A ) t ( l - t ) |J + / J e x p ( - A t ) { ( l - t ) P " (integration  by p a r t s )  1  P  dt.  (6.8.4)  p-1 dt  - (p-l)t(l-t) " } P  2  dt]  = -p  [/J  exp( At)(l-t) "" dt P  1  r  (p-D/5 r  t(l-t)  P  2  exp(-At) d t ] .  P-2  1  E q u a t i o n (6.8.4) thus becomes p(p-l)Jg  t(l-t)  exp(-At) d t .  Consequently, (6.8.3) has the form R(A,A*) = p - [ 2 ( p - l ) / p ] [ p ( p - l ) / J [(p-l) /p][p/J 2  P  2  P  (l-t)  2  = p-2(p-l) /J t ( l - t ) "  t(l-t) " P _ 1  exp (-At) d t ]  2  exp(-At) d t  exp(-At) d t - ' ( p - l ) / J  exp(-At)(l-t)  2  Thus R(A,A*) i s seen to be an i n c r e a s i n g concave f u n c t i o n o f A. summarize the above i n t o the f o l l o w i n g  dt.  p - 1  We  theorem.  Theorem 6.8.5. L e t X^,...,X  be independent P o i s s o n v a r i a b l e s w i t h parameters  A^,...,A , p ^ 2.  Then the r i s k R(A,A*) of the e s t i m a t o r  A* = (1 -  (p-1)/(Z+p-1))X  i s an i n c r e a s i n g concave f u n c t i o n of A =  P E A. from R(0,A*) = 1/p to i-1 1  R(«>,A*) = p.  R  F  A  R ( A  It  X  '  A  * ) = }  I t has power s e r i e s  _ ^P- ) P 1  D P  v  a l s o has the i n t e g r a l  (j+i)r( +i) r(p+j+i) P  . j _ (p-D  ( (  A  ) P  2  r(p+i) ..! r( +j+i)  j •  p  0  (  A  )  P  representation  R(A,A*) = p - ( p - l ) / J e x p ( - A t ) ( l - t ) 2  P _ 1  d t - 2(p-l) /Jexp(-At) t ( l - t ) " 2  P  2  dt.  T h i s r e s u l t i s v e r y s i m i l a r to C o r o l l a r y 2 of E f r o n and M o r r i s [1973], which d e a l s w i t h the normal case.  94  6.9  R i s k F u n c t i o n of A * as a F u n c t i o n of A The next theorem g i v e s the r i s k of the e s t i m a t o r  of X  A * as a f u n c t i o n  (A.,..., X ). 1 p  =  Theorem 6.9.1. The of X =  r i s k of the e s t i m a t o r  X*  =  (1 -  [(p-1)/(Z+p-1)])X  as a f u n c t i o n  ( A T ..... A ) i s 1 p  R(A,A*)  = p -  (p-l) E^  [ l / ( Z + p ) ] - 2 ( p - l ) E ^ [l/{(Z+p)(Z+p-1)}]  2  = p - (p-l) E  2  [(Z+p+l)/{(Z+p)(Z+p-1)}].  2  A  (6.9.2)  Proof: The  l e f t hand s i d e of  (6.9.2),  as p o i n t e d out e a r l i e r ,  P E A . , say f ( A ) . The r i g h t hand s i d e o f i=l a l s o a f u n c t i o n of A, say g ( A ) . By d e f i n i t i o n , f u n c t i o n of A =  is a  (6.9.2) i s  1  E  b  f ( A ) = p-  (p-l) b/p 2  2(p-l)b /p 2  where E^ i n d i c a t e s e x p e c t a t i o n w i t h r e s p e c t to the j o i n t p r i o r bution  E  (6.4.1) w i t h a = p.  g(A)  b  where E  b  2  b  b  (p-1)  2  A  E  b  2  E  [l/(Z+p)] - 2 ( p - l )  2  b  E E  b  A  [1/{ (Z+p) (Z+p-1) } ] [l/{(Z+p)(Z+p-1)}]  i n d i c a t e s e x p e c t a t i o n w i t h r e s p e c t to the m a r g i n a l  of Z ^ NB(b,p+l).  E  Also,  = p - ( p - l ) E E [ l / ( Z + p ) ] - 2 (p-1) = p -  g(A)  = p -  As a r e s u l t ,  (p-l) b/p 2  distri-  2(p-l)b /p. 2  distribution  .95  T h e r e f o r e E, g(A) = E, f ( A ) f o r every v a l u e of b, 0 < b < 1, o r b b ~~ e q u i v a l e n t l y , f o r every v a l u e of a, a > 0. A = A  P  S i n c e the d i s t r i b u t i o n s of  exp(-A/a) dA a r e complete as a f u n c t i o n o f a, f and g must be the  same f u n c t i o n . Q.E.D.  Both Theorem 6.8.5 and Theorem 6.9.1 show t h a t R(A,A*) < p f o r a l l A, so A* i s b e t t e r than the MLE  (whose r i s k i s p) f o r a l l A.  [ l / ( z + p ) ] and [l/{(z+p)(z+p-1)}]  a r e convex f u n c t i o n s i n z.  Furthermore, Hence by  Jensen's i n e q u a l i t y , the upper bound UB(A) of R(A,A*) = R(A,A*) i s of the form  « » - - ^  -x f ^ -  -  _ (p-1)  A +2pA+p -p  2  _  2(p-1) ,A-U^  A+P  2  '  2  (A+p) -p In p a r t i c u l a r , when A = 0, then UB(0) = 1/p.  Since  R(0,A*) = 1/p,  the upper bound i s a t t a i n e d when A = 0.  6.10  R i s k F u n c t i o n of the Clevenson-Zidek  Estimators  By Theorem 2.2.9, the e s t i m a t o r s  A  S  = (1 - [s(p-1)/(Z+p-1)])X,  0 < s < 2,  ~g a r e u n i f o r m l y b e t t e r than the MLE.  The r i s k f u n c t i o n s of A  to those g i v e n i n Theorem 6.8.5 and Theorem 6.9.1.  are similar  They a r e g i v e n i n the  f o l l o w i n g theorems ( p r o o f s a r e s i m i l a r to those i n Theorems 6.8.5 and 6.9.1)  96  Theorem 6..10.1. The  r i s k f u n c t i o n of A  (0 < s < 2) as a f u n c t i o n of b i s  R(b,A ) = p - ( p - l ) b [ l - ( s - l ) ] / p S  2  2b s(p-l)/p.  2  2  Theorem 6.10.2. s r i s k f u n c t i o n of A A  The  R(A,A ) = p S  (p-l) [l  (0 < s < 2) as a f u n c t i o n of A =  - (s-l) ]/J  2  2  exp(-At) ( l - t )  - 2 ( p - l ) s / J exp(-At) t ( l - t ) ~ 2  P  2  P  Z  A_^ i s  dt  P _ 1  dt.  T h i s i s an i n c r e a s i n g concave f u n c t i o n of A from R(0,A ) = 1/p S  +  ( p - l ) ( s - l ) / p + 2 ( p - l ) ( l - s ) / p to R(~,A ) = 2  2  S  p.  Theorem 6.10.3. s r i s k f u n c t i o n of A A  The  (0 < s < 2) as a f u n c t i o n of A = (A.,...,A ) _ _ 1 p  i s R(A,A ) = p - ( p - l ) [ l - C s - 1 ) J E [ 1 / ( Z + p ) ] - 2 ( p - l ) s E S  2  2  2  A  6.11  [ l / { (Z+p) (Z+p-1)}].  Summary In t h i s s e c t i o n , we  form A = (1 - b ( Z ) ) X and  confine  our  a t t e n t i o n to e s t i m a t o r s  c a r r y out our  a n a l y s i s by means of  Bayes methods.  We  normal case and  then proceed to c a l c u l a t e the  (a  A  term c o i n e d  of A of empirical  b e g i n w i t h a d e s c r i p t i o n of'-.the. method used i n  by E f r o n and  Morris  [1973]) i n the P o i s s o n case when The  r e l a t i v e savings  l o s s p l a y s a fundamental r o l e i n the a n a l y s i s performed by E f r o n  and  [1973] i n the normal c a s e .  "truncated The  r i s k R(A,A  the  " r e l a t i v e savings l o s s "  n o r m a l i z e d squared e r r o r l o s s i s the c r i t e r i o n .  Morris  We  obtain "plus r u l e s  (A  =  and  (l-b(Z)) X) +  Bayes r u l e s " , r e s u l t s s i m i l a r to those i n the normal c a s e .  remainder of the s e c t i o n i s devoted to the c a l c u l a t i o n of  g  the  ) of the Clevenson-Zidek  estimators  the  XS  = [1 - s(p-1)/(Z+p-1)]X  as a f u n c t i o n of X, where 0 <_ s <^ 2.  We  f i n d t h a t R(A,A  a c t u a l l y a f u n c t i o n o f A o n l y through the v a l u e  Moreover, R(A,A  g  of A =  ) is  P E i=l  A..  ) i s an i n c r e a s i n g and concave f u n c t i o n of A.  98  SECTION 7.  7.1  COMPUTER SIMULATION  Introduction In t h i s s e c t i o n we d e s c r i b e the r e s u l t s of a computer  simulation  used to compare some o f our proposed e s t i m a t o r s w i t h the MLE.  We  also  compare a C l e v e n s o n - Z i d e k e s t i m a t o r and one of Peng's e s t i m a t o r s (Theorem 2. 2.3)-with the MLE.  F i n a l l y , we  d i s c u s s the performance of the  estimators. The computations r e p o r t e d here were performed both on the IBM computer  370/168  a t the U n i v e r s i t y o f B r i t i s h Columbia and the Data G e n e r a l NOVA  840 computer program was  a t the U n i v e r s i t y of C a l i f o r n i a , R i v e r s i d e . used i n the IBM computer  NOVA computer.  First,  v a r i a b l e s i s chosen.  A FORTRAN  and a BASIC program was  used i n the  the number p o f independent P o i s s o n random Second, p parameters A^ a r e g e n e r a t e d randomly  w i t h i n a c e r t a i n range  (c,d).  T h i r d , one o b s e r v a t i o n of each of the  p d i s t r i b u t i o n s w i t h the parameters o b t a i n e d i n the second step i s generated.  E s t i m a t e s of the parameters a r e then c a l c u l a t e d a c c o r d i n g to  the e s t i m a t o r A we want to t e s t . and the r i s k s under and the MLE  The t h i r d  s t e p i s r e p e a t e d 2000 times  the r e l e v a n t l o s s f u n c t i o n s f o r both the e s t i m a t o r  are c a l c u l a t e d .  as compared to the MLE,  [  R  The p e r c e n t a g e of the s a v i n g s i n u s i n g A (  X  >  R (  X  ~  X  ) *  whole p r o c e s s i s then r e p e a t e d a t l e a s t  1 0 0  ]  %. i s c a l c u l a t e d .  The  t h r e e times and the average  p e r c e n t a g e o f the s a v i n g s i s c a l c u l a t e d . In the case when our B a y e s i a n e s t i m a t o r s hyperparameters  (u,v,y,a,b) which s p e c i f y t h e p r i o r i n f o r m a t i o n a r e chosen  beforehand, i . e . b e f o r e the t h i r d the parameters  ( S e c t i o n 5) a r e used, the  A^ i n such a way  s t e p takes p l a c e .  We  chose the range o f  t h a t we might check the performance o f  99  the e s t i m a t o r s when the parameters X another.  a r e r e l a t i v e l y c l o s e to one  T h i s i s e s p e c i a l l y r e l e v a n t when the performance o f our Bayes  e s t i m a t o r s a r e checked, s i n c e t h i s i s the case where we expect s u b s t a n t i a l improvement over  that a  the u s u a l procedure w i l l take p l a c e .  For  purposes o f comparison, we a l s o i n c l u d e a s i m u l a t i o n study o f the p e r formance o f the e s t i m a t o r s when the range o f the parameters A^ i s wide. Our than  e x p e c t a t i o n t h a t the Bayes e s t i m a t o r might n o t always be b e t t e r the MLE when the range o f the A^'s i s wide was s u b s t a n t i a t e d when  n e g a t i v e s a v i n g s r e s u l t e d from a c h o i c e o f p = 3, u ~ 9.0, and (c,d) =  (0,20).  The MLE performed w e l l i n t h i s case because p i s s m a l l , the  range o f the A / s i s wide, and the p r i o r was p u r p o s e l y  s e l e c t e d t o be  an improper c h o i c e f o r the problem. I n c a l c u l a t i n g t h e p e r c e n t a g e o f improvement o f t h e v a r i o u s over  the MLE, the a p p r o p r i a t e l o s s f u n c t i o n s must be used. P  L  (A,A) =  E x=l  2 k (A. - A.) /X..  estimators  R e c a l l that  I n T a b l e s I and I I , t h e l o s s f u n c t i o n L~ i s  the c r i t e r i o n , w h i l e i n T a b l e s I I I and IV, L^ i s used.  The l o s s  f u n c t i o n L^ i s the c r i t e r i o n i n T a b l e V, and T a b l e s VI through XIV use squared  error loss.  In most o f the cases,  t h e improvement percentage i s seen t o be an  i n c r e a s i n g f u n c t i o n o f p, the number o f independent P o i s s o n  distributions.  For the non-Bayes e s t i m a t o r s , we see t h a t i n g e n e r a l , the improvement percentage decreases  as the magnitude o f the A^'s i n c r e a s e s .  The improve-  ment p e r c e n t a g e o f the Bayes e s t i m a t o r s depends on the c h o i c e o f the p r i o r hyperparameters; proper over  the MLE.  c h o i c e l e a d s to s u b s t a n t i a l improvement  100  7.2  E s t i m a t o r s Under k-NSEL In S e c t i o n 4 we  k estimators A  d e r i v e d , f o r each n o n - n e g a t i v e i n t e g e r k, a c l a s s of  A  P  dominating  the MLE  under k-NSEL L  (A,A)  =  Z  * 2 (A.-A.)  k /A..  1=1 These e s t i m a t o r s a r e of the form (f>(Z)X ( X . - l ) . - - (X.-k+l) A^ = X. X X  1  j^i  X.(X.-l)•••(X.-k+l)  i s nondecreasing.  The  "2 estimator A  i s of c o n s i d e r a b l e a p p e a l because t h i s i s the case where a P X Z (1 - - — ) i s used i n e s t i m a t i n g s c a l e A. . x=l x: computer s i m u l a t i o n r e s u l t s i n t h i s s u b s e c t i o n are m a i n l y  n a t u r a l loss function L (A,A) = 0  2  parameters.  1  J  where <p(z) e [ 0 , 2 k ( p - l ) ] and ( i . e . k=2)  1  p Z (X.+l)---(X.+k) +  Our  ±  2  .  "2 devoted to the performance of an e s t i m a t o r "1 estimator A . f o r A , and A 2  1  More s p e c i f i c a l l y , we = ( 1 ™ -= —)Z.  A  and a  Clevenson-Zidek  choose (p(z) = 2[p-1] ( i . e . k [ p - l ] )  P  z+p In T a b l e I, we  see t h a t f o r the ranges c o n s i d e r e d ,  the percentage  "2 of improvement i n r i s k of A  parameters f a l l  over  the MLE  i n t o a narrow i n t e r v a l .  i s c o n s i d e r a b l e when the  "2  For each v a l u e of p, A  b e s t when the parameters a r e i n the i n t e r v a l s improvement decreases  (0, 4) and  (4, 8 ) .  performs The  g r a d u a l l y as the magnitude of the A / S i n c r e a s e s . "1  In c o n t r a s t , the Clevenson-Zidek  estimator A  the parameters a r e r e l a t i v e l y s m a l l , w i t h  performs v e r y w e l l o n l y when  the improvement p e r c e n t a g e  d e c r e a s i n g d r a m a t i c a l l y as the magnitude of the A _ / s i n c r e a s e s ( T a b l e I I I ) . T h i s i s as c o n j e c t u r e d i n S e c t i o n 4.  T a b l e s I I and  IV show r e s u l t s f o r  "2 wider ranges of the A^'s. over  the MLE  Although  the improvement percentages  a r e by no means s u b s t a n t i a l ,  they a r e n e v e r t h e l e s s  of A greater  101  -Table IV  Improvement Percentage o f A  over the MLE  Narrow Range f o r x.'s  Range o f the  Percentage o f Improvement over the MLE  Parameters X.  1  p=3  p=4  p=5  p=8  (0, 4)  24.97  23.53  34.35  33.94  35 .43  (4, 8)  23.08  25.93  28.83  30.95  32 .53  (8, .12)  14.89  19.60  20.69  21.58  24 .12  (12, 16)  11.04  12.53  15.33  17.92  18 .09  Table I I .  Improvement Percentage o f X  over the MLE  Wide Range f o r A ' s  Range o f the  Percentage o f Improvement  Parameters X. l  over the MLE  (0, 20)  11.62  6.59  (10,30)  11.99  12.01  =10  102  Table I I I .  Improvement Percentage o f A  over the MLE  Narrow Range f o r A.'s  Range o f the  Percentage o f Improvement over the MLE  Parameters A.  P=3  p=4  p=5  p=8  p=10  (0, 4)  24.77  26.70  27.97  29.47  29.89  (4, 8)  7.34  8.29  9.83  11.21  12.46  (8, 12)  4.02  5.85  6.05  7.00  7.69  (12, 16)  2.44  2.74  4.28  5.36  5.18  X  T a b l e IV.  Improvement Percentage o f A"*" over the MLE Wide Range f o r A^'s  Range o f the  Percentage o f Improvement  Parameters A.  over the MLE  p=5  p=8  (0, 20)  .8.23  6.87  (10, 30)  4.08  4.32  103  "1 than those o f X .  Of course,  the d i f f e r e n t l o s s f u n c t i o n s employed f o r  "2 "\ X and X might c o n t r i b u t e t o such a d i f f e r e n c e .  -4 T a b l e V r e v e a l s the performance of the e s t i m a t o r <p(z) = 4[p-rl] ( i . e . k [ p - l ] ) . interval 30%,  which i s r a t h e r s u b s t a n t i a l .  expected  o f improvement a r e seen to be above However, when the parameters a r e i n  (0, 4 ) , the improvements a r e u n i m p r e s s i v e .  because X  with  When the parameters a r e c o n f i n e d t o the  (4, 8 ) , the percentages  the i n t e r v a l  X  gives i d e n t i c a l estimates  T h i s i s as  as the MLE when the  o b s e r v a t i o n s a r e l e s s than f o u r , and when the X.'s a r e i n the i n t e r v a l l (0, 4 ) , such o b s e r v a t i o n s a r e l i k e l y the p r o o f o f Theorem 4.3.1).  to occur  ( c f . the remarks f o l l o w i n g  The s l i g h t improvements shown i n the  (0, 4) row o f T a b l e V a r e due t o the f a c t t h a t some o b s e r v a t i o n s a r e g r e a t e r than f o u r .  While the MLE e s t i m a t e s X^ by x^, the e s t i m a t o r X  s h r i n k s the o b s e r v a t i o n s  g r e a t e r than f o u r towards z e r o , r e s u l t i n g i n  some improvement. 7.3  Bayes The  Estimators  Bayes e s t i m a t o r s X developed  i n S e c t i o n 5 a r e o f the form  X. = E . X . l X,a,b,u i v  Z(X +1) ±  =  ~(Z+F)  g(X+e ,a,b,u) ±  g(X,a,b,p)  '  1  =  1  "'-»P  ( 7  - ' 3  x )  where g(x,a,b,y) i s t h e g e n e r a l i z e d hypergeometric f u n c t i o n g i v e n i n equation  (5.2.4).  Basically,  (7.3.1) depends on the c h o i c e o f the p r i o r  d i s t r i b u t i o n o f a, which i s g i v e n i n e q u a t i o n  (5.2.3).  A particular  p r i o r d i s t r i b u t i o n o f a i s determined by the s p e c i f i c a t i o n o f the  104  T a b l e V.  Improvement Percentage o f A  Range of the  over the MLE  Percentage o f Improvement over the MLE  Parameters X. p=4  p=5  p=8  p=10  (0, 4)  7.51  0.57  6.56  3.80  (4, 8)  30.03  34.23  38.25  31.89  1  Table VI.  Poisson-Distributed  a (u=0, v=0)  Narrow Range f o r A.'s  Range of the  Percentage o f Improvement over the MLE  Parameters A. V  p=3  p=4  p=5  p=8  p=10  (0, 2)  0.5  47.81  42.33  54.52  60.17  54.25  (2, 4)  2.0  45.29  50.19  54.87  63.03  65.01  (4, 8)  5.0  46.02  47.00  57.94  58.74  61.28  (8, 12)  9.0  43.66  52.33  58.70  61.93  66.65  (12, 16)  13.0  47.18  55.57  57.35  65.92  65.00  (16, 20)  17.0  46.71  53.13  55.25  64.90  65.03  105  hyperparameters (u,v,a,b,y). ourselves  The  to the f o l l o w i n g  In our  s i m u l a t i o n study, we  cases;  0 <_ \i < <x> ( P o i s s o n - d i s t r i b u t e d  v=0,  shallj.restrict  (1)  u=0,  a)  (2)  u=l, v=0,  a^=1.0, 0 <_ y < 1 ( g e o m e t r i c - d i s t r i b u t e d  (3)  u=l, v=0,  a^=3.0, 0 <_ y < 1 (negative b i n o m i a l - d i s t r i b u t e d  a)  c h o i c e of y r e f l e c t s one's b e l i e f about the magnitude of the  meters A_^.  A.'s l  para-  A r e l a t i v e l y l a r g e v a l u e of y i n d i c a t e s t h a t the A^'s  b e l i e v e d to be l a r g e , w h i l e a r e l a t i v e l y s m a l l v a l u e means t h a t are b e l i e v e d to be  the  Observe t h a t f o r g i v e n y and  3»  the  of X_^ i s  E(X.|S) = E E(X.|(3,A) = E(A.|3) = E E(A. | a, 3) = E ( f | 3) = \  1  l  are  small.  R e c a l l t h a t A_^ ^ gamma (a, 3 ) . marginal expectation  a).  where the e x p e c t a t i o n s  1  x;  p  are taken a p p r o p r i a t e l y .  be c l o s e to the parameter A^,  p  Note t h a t Ea i s a  f u n c t i o n of y. Suppose t h a t the s c a l e parameter 3 i s e q u a l to one, problem i s c a s t i n terms of u n i t s of 3.  Ea  In order  so t h a t  our  t h a t E(X_j3=l) might  y i s chosen so t h a t E^A  = E(X_j3=l) =  c + d — - —  i f the A 's a r e b e l i e v e d to be i n the i n t e r v a l  example, i f ( c , d) = (4, 8 ) , y=5 way  to choose y; other  three values the g r e a t e s t  of y were chosen.  However, t h i s i s o n l y  are c e r t a i n l y p o s s i b l e . The  to suspect  y than t h a t d e s c r i b e d  c h o i c e of y=12  one  In Table V I I ,  seemed to r e s u l t  in  above.  t h a t t h e r e are i n f a c t b e t t e r c h o i c e s In r e t r o s p e c t , i t seems t h a t we  not o n l y the l e n g t h of the i n t e r v a l as a measure of  but a l s o , and  For  savings.  T h i s l e a d s us  consider  choices  i s chosen.  (c, d ) .  p o s s i b l y more i m p o r t a n t l y ,  for  should closeness,  the r e l a t i v e magnitude of  the  106  A^'s w i t h i n an i n t e r v a l .  We  might use  the r a t i o c/d, where c > 0, as  i n d i c a t o r of the c l o s e n e s s of the A.'s.  1  the A^'s may  When the r a t i o i s c l o s e to z e r o ,  be s a i d to be r e l a t i v e l y spread out i n t h a t i n t e r v a l ,  when the r a t i o i s c l o s e to one,  they may  an  be s a i d  to be r e l a t i v e l y  Hence f o r i n t e r v a l s of f i x e d l e n g t h , the parameters  and close.  are considered  to  be c l o s e r i f the lower bound of the i n t e r v a l i s f u r t h e r away from z e r o . In o t h e r words, the c l o s e n e s s of the A^'s  does not depend s o l e l y on  E u c l i d e a n d i s t a n c e between the A.'s, but a l s o on t h e i r  the  location.  I  R e c a l l t h a t the Bayes e s t i m a t e i s a weighted average of x^ and If  the A^'s  to  x, which means t h a t y should be g i v e n a r e l a t i v e l y l a r g e v a l u e .  the A / s  a r e v e r y c l o s e to one another, more weight should be  a r e spread out, a s m a l l v a l u e of y should be chosen.  T a b l e V I I I , we  see t h a t a c h o i c e of y=0.1  the i n t e r v a l i s (0, 20).  i s b e t t e r than 1.0  close  to  However, i n the i n t e r v a l  s u p e r i o r to the MLE i f we  (10, 30), the  i s b e t t e r than a c h o i c e of y=14,  t h a t encountered  If  In and 9.0  i n Table VII.  when the  i . e . a s m a l l v a l u e of  ( a l t h o u g h the l e n g t h of the i n t e r v a l i s s t i l l  a c h o i c e o f y=19  given  S i n c e the parameters a r e q u i t e spread out,  Bayes e s t i m a t e s h o u l d g i v e more weight to the MLE, y i s suitable.  x.  20)  A 's. a r e and we  relatively see t h a t  a phenomenon s i m i l a r  Thus, the Bayes e s t i m a t o r s remain  even though the range of the parameters A  i s wide,  a r e a c u t e enough to s e l e c t the hyperparameters a p p r o p r i a t e l y . From T a b l e VI, we  the MLE  conclude  i s s u b s t a n t i a l when the parameters f a l l  which i s as expected.  of improvement  over  i n t o a narrow i n t e r v a l ,  T a b l e s IX and X both show t h a t the Bayes e s t i m a t o r s  u s i n g other p r i o r d i s t r i b u t i o n s d i s t r i b u t e d a) a r e s t i l l squared  t h a t the percentage  ( n e g a t i v e b i n o m i a l - and  s u p e r i o r to the MLE  error i s substantial.  and  geometric-  the s a v i n g s i n mean  107  Table VII.  Poisson-Distributed  a (u=0, v=0>  E f f e c t of y  Range o f the  Percentage o f Improvement over the MLE  Parameters A. v  p=4  p=5  p=8  (4, 8)  1.0  29.93  35.51  47.57  44.47  (4, 8)  5.0  47.00  57.94  58.74  61.28  (4, 8)  12.0  60.69  64.91  65.94  72.04  1  Table V I I I .  Poisson-Distributed  a (u=0, v=0)  Wide Range f o r ^ ' s A  Range o f the  Percentage o f Improvement  „ , Parameters A .  over the.MLE  Vi  P=5  . p=8  (0, 10)  2.0  8.57  29.38  (0, 20)  0.1  9.34  15.84  (0, 20)  1.1.0  7.85  12.12  (0, 20)  9.0  9.12  3.05  (10, 20)  14.0  48.51  52.92  (10, 30)  14.0  11.45  10.90  (10, 30)  19.0  33.43  30.21  p=10  108  T a b l e IX.  N e g a t i v e B i n o m i a l - D i s t r i b u t e d ci (u=l, v=0, a^=3.0)  Range o f the  Percentage o f Improvement over the MLE  Parameters A.  1  y  p=3  p=4  p=5  p=8  p=10  (0, 4)  0.25  41.99  55.73  53.71  44.66  47.44  (4, 8)  0.5  42.60  47.11  48.35  57.08  55.61  (8, 12)  0.75  44.61  49.70  55.83  61.68  65.41  T a b l e X.  G e o m e t r i c - D i s t r i b u t e d a ( u = l , v=0, a^=1.0)  Range o f the  Percentage o f Improvement over the MLE  Parameters A. y  p=3  p=4  p=5  p=8  p=10  (0, 4)  0.5  35.17  34.69  37.08  47.77  51.58  (4, 8)  0.833  39.20  47.42  46.68  60.84  59.16  109  7.4  E s t i m a t o r s Under Squared E r r o r  Loss  " (0) We s h a l l next compare the performance of Peng's e s t i m a t o r A (a s p e c i a l case o f A ^ ^ d e s c r i b e d i n S e c t i o n 3) and our new e s t i m a t o r A ^ , which s h r i n k s the MLE towards the minimum o f the o b s e r v a t i o n s , where M™1  = x.  A[ 1  1  (p-N -2) H (X) m  + 1  , 1 - 1  p,  p •S H?(X) 1=1 9  1  and  the H^'s a r e as d e f i n e d i n e q u a t i o n  t h i s adaptive estimator A ^ broad  spectrum  M  (3.4.1).  We have argued  that :  ^ should perform b e t t e r than A ^ ^ a c r o s s a  o f v a l u e s f o r the A . ' s . I  T a b l e XI shows t h a t a l t h o u g h A ^ ^ p r o v i d e s a n o t i c e a b l e improvement over  the MLE, the improvement percentage  move away from  zero.  decreases r a p i d l y as the A ^ ' s  I n c o n t r a s t , the improvement percentages  X I I I remain n o t i c e a b l e even when t h e A_^'s a r e i n t h e i n t e r v a l T h i s s u p p o r t s our c o n j e c t u r e t h a t the e s t i m a t o r A when p ^ 4.  The improvement percentages  i n Table (12, 1 6 ) .  ^ i s s u p e r i o r to A ^ ^  f o r p=3 i n T a b l e X I I I a r e a l l  Tm1 zero because A  i s i d e n t i c a l w i t h the MLE when p <^ 3.  Thus Peng's  e s t i m a t o r has the m e r i t t h a t i t p r o v i d e s a b e t t e r e s t i m a t o r than the MLE under squared  e r r o r l o s s when p >_ 3.  Note, however, t h a t use o f A  ^  i m p l i c i t l y i n v o l v e s a c h o i c e o f k=0, towards which the o b s e r v a t i o n s a r e shrunk.  I f t h i s k i n d of s u b j e c t i v i t y i s to avoided and the s h r i n k a g e ~ Tml  determined and  o n l y by the d a t a  improvement over  (using A  ) , one degree o f freedom i s l o s t  the MLE r e s u l t s o n l y when p ^ 4.  F i n a l l y , T a b l e s X I I and XIV show our r e s u l t s f o r wide ranges parameters A_^.  I n both c a s e s , the improvement percentages  o f the  a r e seen t o be  minimal, which i s n o t s u r p r i s i n g because the r i s k o f the MLE i s l a r g e i n t h i s case, and hence the r e l a t i v e s a v i n g s a r e bound t o be s m a l l .  110  Table XI.  Improvement Percentage of A  over the MLE  Narrow Range f o r A.'s  Range of the  Percentage o f Improvement over the MLE  Parameters A.  1  p=3  P=4  p=5  p=8  p=10  (0, 4)  2.23  3.97  6.37  7.70  8.52  (4, 8)  0.58  1.14  1.42  1.68  1.79  (8, 12)  0.17  0.36  0.50  0.63  0.85  (12, 16)  0.05  0.00  0.19  0.37  0.48  Table X I I .  Improvement Percentage o f A ^  over the MLE  Wide Range f o r A^'s  Range of the  Percentage of Improvement  Parameters A. l  over the MLE  P=5  p=8  (0, 20)  0.69  0.62  (10, 30)  0.37  0.40  Ill  Table XIII.  Improvement Percentage of A  over the MLE  Narrow Range f o r A.'s  Range of the  Percentage of Improvement over the MLE  Parameters A.  1  p=3  p=4  p=5  (0, 4)  0.00  4.21  (4, 8)  0.00  (8, 12) (12, 16)  T a b l e XIV.  p=8  p=10  6.35  12.41  11.76  3.80  6.08  .7.95  7.69  0.00  3.81  5.50  6.37  6.59  0.00  2.95  4.74  5.90  5.22  Improvement Percentage of A  over the MLE  Wide Range f o r A.'s  Range of the Parameters A. l  Percentage o f Improvement over the MLE  p=5  p=8  (0, 20) .  0.79  0.68  (10, :30)  2.30  1.85  112  7.5  Comparison o f the E s t i m a t o r s Based! on our computer  estimators considered,  s i m u l a t i o n r e s u l t s , i t appears t h a t among the  the Bayes e s t i m a t o r s p r o v i d e by f a r the most  improvement over the MLE when the parameters a r e i n a . r e l a t i v e l y interval.  narrow  When p r i o r knowledge i n d i c a t e s t h a t the A^'s a r e exchangeable  and a r e l i k e l y  to f a l l i n t o a c e r t a i n r e l a t i v e l y narrow i n t e r v a l  t h e r e i s no doubt t h a t the Bayes e s t i m a t o r s h o u l d be used.  (c, d),  Recall also  t h a t the Bayes e s t i m a t o r s remain s u p e r i o r t o t h e MLE even though the range o f the parameters A^ i s wide, i f we choose the hyperparameters suitably.  I t s h o u l d be n o t e d , however, t h a t the Bayes e s t i m a t o r s do  not dominate  the MLE u n i f o r m l y i n A, and hence should n o t be used  indiscriminately. On the other hand, t h e e s t i m a t o r s A^, A ^ ,  and A ^  a r e guaranteed  to have lower r i s k than the MLE u n i f o r m l y i n A under a p p r o p r i a t e functions.  loss  ~k "(k) The e s t i m a t o r s A and A can be used advantageously when  the i n t e g e r k i s chosen a p p r o p r i a t e l y .  The e s t i m a t o r A ^  p •> 4 and p r i o r knowledge o f k used i n the e s t i m a t o r  is.useful i f  " (k) A i s vague.  113  SECTION 8.  8.1  EXTENSIONS  Motivation Up  to t h i s p o i n t , we  have c o n f i n e d our a t t e n t i o n to the s i t u a t i o n i n  which o n l y one o b s e r v a t i o n i s taken from each of p independent populations.  We  now  suppose t h a t X ^,...,X.  ^ Poisson  ( A . ) , where  i n i  x n  ;L.!•» i  n. i - = ^ J-i=1  X  X  following  Question  1»."»P>  =  J  and  t h a t a l l the X_^  , i = l,...,p,  the MLE  x  's a r e independent.  X. X of A i s (—,... T 1 P n  .  n  We  Letting  pose  the  question:  8.1.1.  Given  the s i t u a t i o n d e s c r i b e d above, a r e t h e r e e s t i m a t o r s  which dominate the MLE  suppose t h a t X  ^  The v a r i a b l e s X^/n^  e s t i m a t i n g p normal means 0.,  N(0^,1) f o r i = l , . . . , p and j =  a r e s t i l l normal and we  the problem i n which one  We  t h i s problem.  we  l,...,n^.  are e s s e n t i a l l y d e a l i n g w i t h  o b s e r v a t i o n i s taken from each p o p u l a t i o n .  our case, u n f o r t u n a t e l y , the d i s t r i b u t i o n of X^/n_^ I i f n_^ > 1.  A of A  under an a p p r o p r i a t e l o s s f u n c t i o n ?  In the case of s i m u l t a n e o u s l y independent  s  n  o  longer  However, i f our  i n t e r e s t i s to e s t i m a t e n A^y  In  Poisson  t h e r e f o r e cannot a p p l y our p r e v i o u s r e s u l t s d i r e c t l y  then the f o r e g o i n g t h e o r y can be a p p l i e d because X^ ^ P o i s s o n this  Poisson  i =  to  l,...,p,  (n^A^) i n  case. We  first  c o n s i d e r the squared  f i n d an e s t i m a t o r  error loss function.  A of A such t h a t  P . E, Z (A.-A.) A. . . i i i=l 2  P . X < E, Z (A. - — ) — A., i n . i=l I  2  for a l l A  The  g o a l i s to  114  with s t r i c t  E. A.  and  i n e q u a l i t y f o r some A. P  X  Z  (—  i=l  n  .  l  p = E, Z ±r (X. A .... 2 1 i = l n. l  X.y  -  i  t h a t X. ^ P o i s s o n I  Observe t h a t  (n.A.), i i  n.A.r i  i = l,...,p.  i  We  a r e thus l e d to c o n s i d e r -  a t i o n of the f o l l o w i n g problem.  Problem 8.1.2. Suppose X_^  independent ^ Poisson  o b s e r v a t i o n of X_^,  i = l,...,p.  such t h a t A dominates the MLE  (A^) , i = l , . . . , p , and  I f p o s s i b l e , f i n d an e s t i m a t o r  X = (X^,...,X ) u n i f o r m l y under  g e n e r a l i z e d squared e r r o r l o s s f u n c t i o n L i  t h a t x^  c  P  "  P  =  i s an  A of A.  the  2  Z c.(A.-A.) , where c. > 0, 1 111 1 i=l a  = 1,... , p. S o l u t i o n of Problem 8.1.2  Question  8.1.1.  Problem 8.1.2 8.2  w i l l a u t o m a t i c a l l y p r o v i d e an answer to  In the next s u b s e c t i o n we  e x i s t s provided  s h a l l show t h a t a s o l u t i o n to  c e r t a i n c o n d i t i o n s on the c^'s  E s t i m a t o r s Under G e n e r a l i z e d Squared E r r o r Loss independent L e t X. ^ P o i s s o n ( A . ) , i = l , . . . , p and l l r  The  estimators  A of A we  hold.  X = (X,,...,X ) . 1 p  c o n s i d e r here a r e a g a i n of the form X +  where f ( X ) i s as d e s c r i b e d i n S e c t i o n 2.1  and  f(X)  s a t i s f i e s the f o l l o w i n g  conditions:  The  (1)  f ^ ( x ) = 0 i f x has  (2)  E  a negative  | f . ( X + j e . ) | < - f o r j = 0,  x  coordinate 1.  r e s u l t s i n t h i s s u b s e c t i o n a r e v e r y s i m i l a r to those  a r e the d e r i v a t i o n s .  i n S e c t i o n 6,  The next lemma, which i s s i m i l a r to Lemma 2.2.2,  g i v e s the d e t e r i o r a t i o n i n r i s k of .A = X + f ( X ) as compared to the of the MLE  X.  risk  as  115  Lemma .:8.:2.1. c  "  Under the l o s s f u n c t i o n L (A,A) =  P  Z  " 2 c.(A.-A.)",  the d e t e r i o r a t i o n , i n  1=1 r i s k D o f A = X + f ( X ) as compared to the r i s k o f X i s A,  D = R(A,A) - R(A,X) = E. A where A  A =  P Z c.f.(X) + 2 . .. i i 2  1=1  The f o l l o w i n g  p  Z c.X.ff.CX) - f . ( X - e . ) ] . i l l . . l l i=l theorem g i v e s e s t i m a t o r s A of A  dominating  X under  L . C  Theorem 8.2.2. independent L e t X_^ Poisson (p > 3 ) .  ( A ^ ) , i = l , . . . , p and X = (X^,.. . ,X^)  Define  f.(x) i v  = - —  v  (. Z ,„ /cT - 2v/c*") ,h(x.)/S j : x fO 2 + i  i  i f x. > 0 i -  3  =0  i f x. < 0,  l i ,=! 1,... ,p, where  (1)  P c* = Max{c.} i=l 1  1 Z ^ ifj > 1 n=l j  (2)  h(j) =  = 0  ifj < 0  2 Z h (x.). i=l P  (3)  S =  1  L e t f ( X ) = ( f . ( X ) , . . . , f ( X ) ) . Then the e s t i m a t o r A = X + f ( X ) 1 p c dominates X u n i f o r m l y i n A under the l o s s f u n c t i o n L provided Z i=l  /cT > 2/c*. J  116  Proof: The  proof  i s very  that the A given -(  E j:x#>  s i m i l a r t o t h a t o f Theorem 6.3.8.  i n Lemma 8.2.1 i s l e s s than o r e q u a l t o  /c7 - 2/c*) _/S and hence R(A,A) <_ R(A,X) f o r a l l A. 2  2  Q.E.D.  3  I f we i n t e r p r e t c. as the c o s t o f m i s e s t i m a t i o n P  unit,  I t can be shown  of  length,  then  the c o n d i t i o n  E  ••/—  o f A. p e r squared  r -  / c ^ > 2/c* c a n be i n t e r p r e t e d  as meaning t h a t no c o s t per u n i t l e n g t h o f a p a r t i c u l a r A^ c a n be greater  than t h e t o t a l c o s t p e r u n i t l e n g t h o f the r e m a i n i n g A_^'s.  I n t u i t i v e l y , we would n o t expect t h a t an improvement over X can be made if  t h e r e i s a p a r t i c u l a r c o s t /c7 (per u n i t l e n g t h ) dominating the r e s t  of the c o s t s , s i n c e we know t h a t i n t h e one-dimensional case ( p = l ) , X i s P a d m i s s i b l e under squared e r r o r l o s s . Hence the c o n d i t i o n E /cT > l/c* i=l 1  is intuitively The  reasonable.  estimator  problem 8.1.2. can be o b t a i n e d .  A given  i n Theorem 8.2.2 i s n o t t h e o n l y s o l u t i o n to  I n f a c t , s o l u t i o n s s i m i l a r t o those g i v e n i n S e c t i o n 6 F o r example, we may suppose t h e f u n c t i o n h s a t i s f i e s  Lemma 6.2.2 and o b t a i n theorems s i m i l a r t o Theorems 6.3.8, 6.3.12, 6.3.13, etc.  However, we w i l l n o t c a r r y out t h e d e t a i l s h e r e . 2 L e t c ^ = 1/ru.  The f o l l o w i n g c o r o l l a r y suggests an answer t o  Q u e s t i o n 8.1.1. C o r o l l a r y 8.2.3. independent Let X „ L e t X. = 1  ^  Poisson  n. E X.., i = 1 , — , p . j = i !J  (A_^) where Define  >_ 1, i = l , . . . p , and p >_ 3.  A = (A.,...,A ) as 1 p  5  117  X.  i  S I j:X.^O n-. (  X.  v  n. P j=l E  n  I '  J  < ? n=l Z  1  1  »*-*»P  n  n. = Min{n.} x 1=1 i  0  (2)  ^  X.  1  where (1)  j  _2_) n '+  F, n=l  = 0.  1 2 £ — > — . 1=1 i ' * P  Suppose  n  (8.2.4)  n  X X Then the e s t i m a t o r A dominates the MLE = ( — ,. .., —P-) under the n, n 1 P ^ P ^ 2 e r r o r l o s s f u n c t i o n L(X, X) = £ (X. - X.) . ±  squared  1=1 Remarks: (1)  C o n d i t i o n (8.2.4) guarantees t h a t the proposed e s t i m a t o r  d i f f e r e n t from the MLE w i t h p o s i t i v e (2) A  probability.  When n^ = 1 f o r a l l i , the e s t i m a t o r X i s Peng's  estimator  •  (0)  A  X is  (cf.  Theorem 2.2.3).  Moreover, c o n d i t i o n (8.2.4) h o l d s a u t o m a t i c a l l y  i f p > 3. th (3)  Suppose the i  A  p o p u l a t i o n has sample s i z e n^.  (8.2.4) says t h a t improvement over  the MLE  Condition  i s p o s s i b l e under squared  error  l o s s i f the sample s i z e s n^ o f the other p o p u l a t i o n s a r e n o t too' l a r g e . 8.3  E s t i m a t o r s Under G e n e r a l i z e d k-NSEL Answers to Q u e s t i o n  well.  The techniques  8.1.1  a r e p o s s i b l e under o t h e r l o s s f u n c t i o n s as  a r e much the same as those used above, i . e .  118  Q u e s t i o n 8.1.1 i s transformed i n t o a problem s i m i l a r t o 8.1.2, w i t h a d i f f e r e n t l o s s f u n c t i o n , and a n a l y s i s s i m i l a r to t h a t i n a p r e v i o u s i s employed to d e r i v e our r e s u l t s .  We w i l l t h e r e f o r e  section  f o r the most p a r t  merely s t a t e t h e r e s u l t s . The  f o l l o w i n g lemma i s s i m i l a r to Lemma 2.2.7.  Lemma 8.3.1. L e t X. l i  independent 'v. P o i s s o n (A.), i = l , . . . , p and l e t f . : J I  = l , . . . , p s a t i s f y the conditions  A = X + f(X).  ::,  I  given  Then, under t h e l o s s  c L. (A,A)  F. c.(A. i=l  -> R,  Define  function  - 2 k - A.) /A. ,  p  =  i n Lemma 2.2.6.  P  (8.3.2)  the d e t e r i o r a t i o n i n .risk" of : A as compared t o X i s R(A,A) - R(A,X) =  A, where k  A  A,  =  The  f.(X+ke.) c. - ± + 2  1=1  k  Proof:  p E  1  (x. k)  p Z  c, (X +k) 4  i=l  ( k )  +  1  f.(X+ke.) - f . ( X + ( k - l ) e . ) i 1  1  1  (x.+k)  1  (k)  The p r o o f i s v i r t u a l l y the same as t h a t o f Lemma 2.2.7. next theorem s u p p l i e s e s t i m a t o r s  loss function  t h a t dominate the MLE under the  (8.3.2).  Theorem 8.3.3. independent a. L e t X^ P o i s s o n ( A ^ ) , i = l , . . . , p , and t h e l o s s be  as g i v e n  i n (8.3.2).  Define  f . ( x ) = -k(p-i)/cT7F7 x f / ( s + x f ) k )  i  = 0  l  i  l  k )  i f x. > 0 l  i f x. < 0  l  function  119  i  = 1,...,p, where  (1)  = Min{c.} 1=1 1  (2)  S  1  =  E  (x.+k)  j*i  (3)  Let  x?  k )  1  ( k )  3  = x.(x.-l)(x.-2)--.(x.-k+l).  1 1  X  X  f ( X ) = ( f ( X ) , . . . , f (X)) and A = X + f ( X ) . J_ p  Then the e s t i m a t o r A o f  A dominates the MLE X u n i f o r m l y i n A under the l o s s f u n c t i o n (8.3.2).  Proof: U s i n g s i m i l a r t e c h n i q u e s as i n the p r o o f o f Theorem 4.3.1, we can show t h a t  g i v e n i n Lemma 8.3.1 above s a t i s f i e s  2  c^k (p-l) /  A  2 Z  <  < 0. E i=l  (x.+k)  W  r  Q.E.D. The  f o l l o w i n g c o r o l l a r i e s p r o v i d e o t h e r answers to Q u e s t i o n 8.1.1.  C o r o l l a r y 8.3.4. independent Let  X. = 1 j  X  ^  Poisson  n. E X.., i = 1,...,p. i ^ =  X  = i  h.  _ k(P-D n. n.  x  x  ( A ^ , i = l , . . . , p , j = 1,...,^.  D e f i n e A = (A ,...,A ) by 1 p ((k-2)/2) n  . x^ /(s +x^ ) k )  n  1  k )  Let  i2o;  * i  = l , . . . , p , where n  P. = Max{n.}. i=l  Then A dominates the  MLE  1  X. (  X  n1  n  Proof:  ) under k-NSEL L k  p  Use Theorem 8.3.3  i f k = 1 or  *• (A,A)  P = Z . , i=l  w i t h c. = n. I  9  A  (A. - A.) l I  lr /A. w i t h k = 1 or 2, x  * k-2  k-2  and n o t e t h a t c. = (n ) *  X  2.  C o r o l l a r y 8.3.5. independent Let X „  X. =  1  n. Z  ^  X..,  i = l,...,p.  i  , h  Let  D e f i n e A = (A.,...,A ) by  i i  \  1  p  ((k-2)/2) _  n.  k(p-D n.  x i  Poisson (A^), i = l , . . . , p , j = l , . . . , n ^ .  x^/CsVxf0),  n.  x  .  x  p = l , . . . , p , where n,. = Min{n.}. . -,  Then A dominates the MLE  1=1  X X (—\ .. .,—P-) n n 1 p  under k-NSEL w i t h k > 3.  Observe t h a t c.  Proof:  X  =  ,k-2 (n.) ' X'  , and  ., k-2 = (n ) v  A  if k  There a r e , of c o u r s e , other e s t i m a t o r s dominating the l o s s f u n c t i o n L(A,A) =  P Z c.(A. - A.) i=l  2  k /A..  s i m i l a r to those d e r i v e d i n S e c t i o n 4, and we down the d e t a i l s 8.4  An  13, the MLE  under  The r e s u l t s w i l l  be  s h a l l t h e r e f o r e not s e t  here.  Application  There a r e o t h e r s i t u a t i o n s i n which the r e s u l t s of Theorem 8.2.2 8.3.3  might be u s e f u l . Let P ^ ( A ^ ) ,  1  =  One  such s i t u a t i o n i s d e s c r i b e d as f o l l o w s :  1> •••»?» be p independent  Poisson processes with  and  121  i n t e n s i t y parameters A .  Let  be the number o f counts o f the p r o c e s s  observed  d u r i n g the p e r i o d o f time ( 0 , t . ) , i = l , . . . , p . The MLE o f ' • X X A = (A-,..., A ) i s (—,••• ,—2-) • The r e s u l t s i n s u b s e c t i o n 8.2 and 8.3 1 p 1  P  show t h a t t h e r e a r e other e s t i m a t o r s A o f A dominating squared  error  conditions.  l o s s o r under k-NSEL, p r o v i d e d  the MLE under  the t ^ ' s s a t i s f y  I n f a c t , the e s t i m a t o r s d e s c r i b e d i n C o r o l l a r i e s  certain 8.2.3,  8.3.4, and 8.3.5 can be employed here w i t h n^ r e p l a c e d by t ^ and n. = t . = M i n i t . } and n * * i  = t  = Max{t.}. i  122  SECTION 9.  SUGGESTIONS FOR FURTHER RESEARCH  In S e c t i o n 6, we gave an e m p i r i c a l Bayes i n t e r p r e t a t i o n f o r t h e e s t i m a t o r s d e r i v e d under the n o r m a l i z e d  squared  e r r o r l o s s L^.  It is  t h e r e f o r e n a t u r a l to seek a s i m i l a r i n t e r p r e t a t i o n f o r our new e s t i m a t o r s ~k A i n S e c t i o n 4 which a r e d e r i v e d under k-NSEL L, , w i t h k > 2. Moreover, k — s i n c e some o f the e s t i m a t o r s under a r e a c t u a l l y Bayes e s t i m a t o r s , one "k might attempt t o check i f some o f the e s t i m a t o r s o f A e s t i m a t o r s under some a p p r o p r i a t e p r i o r d i s t r i b u t i o n s .  a r e Bayes I f such p r i o r  d i s t r i b u t i o n s can be found, then we have a c l e a r u n d e r s t a n d i n g of when ~k we should use those e s t i m a t o r s A . In the squared e r r o r l o s s case, Peng [1975] has a l r e a d y suggested " (0) t h a t some i n s i g h t s about h i s e s t i m a t o r s A be o b t a i n e d  (see Theorem 2.2.3) might  from an e m p i r i c a l Bayes i n t e r p r e t a t i o n f o r the e s t i m a t o r s .  Up to t h i s p o i n t , no s u c c e s s f u l attempt has been made. " (k) Though the e s t i m a t o r s A under squared be o f i n t e r e s t  suggested  i n S e c t i o n 3 dominate the MLE  e r r o r l o s s when p > 3, they a r e n o t a d m i s s i b l e .  I t might  to see i f t h e r e a r e a d m i s s i b l e e s t i m a t o r s which a l s o  dominate the MLE.  123  BIBLIOGRAPHY  1.  Alam, K. [1973], "A F a m i l y of A d m i s s i b l e Minimax E s t i m a t o r s of the Mean o f a M u l t i v a r i a t e Normal D i s t r i b u t i o n , " Annals o f S t a t i s t i c s , 1, 517-525.  2.  Alam, K. [1975], "Minimax and A d m i s s i b l e Minimax E s t i m a t o r s of the Mean of a M u l t i v a r i a t e Normal D i s t r i b u t i o n f o r Unknown C o v a r i a n c e M a t r i x , " J o u r n a l o f M u l t i v a r i a t e A n a l y s i s , 5, 83-95.  3.  Abramowitz, M., and Stegun, I . [1964], Handbook of M a t h e m a t i c a l F u n c t i o n s , N a t i o n a l Bureau of Standards, A p p l i e d Mathematics S e r i e s , 55.  4.  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[1975], "Simultaneous E s t i m a t i o n o f the Parameters o f Independent P o i s s o n D i s t r i b u t i o n s , " T e c h n i c a l Report No. 78, Department o f S t a t i s t i c s , S t a n f o r d U n i v e r s i t y .  28.  R a i n v i l l e , E. D. [1960], S p e c i a l F u n c t i o n s , New York:  29.  S t e i n , C. [1956], " I n a d m i s s i b i l i t y o f the U s u a l E s t i m a t o r f o r the Mean o f a M u l t i v a r i a t e Normal D i s t r i b u t i o n , " i n P r o c e e d i n g s o f the T h i r d B e r k e l e y Symposium, pp. 197-206, B e r k e l e y : U n i v e r s i t y o f C a l i f o r n i a Press.  30.  S t e i n , C. [1974], " E s t i m a t i o n o f the Parameters of a M u l t i v a r i a t e Normal D i s t r i b u t i o n I : E s t i m a t i o n o f the Means," T e c h n i c a l Report No. 63, Department o f S t a t i s t i c s , S t a n f o r d U n i v e r s i t y .  31.  Strawderman, W. [1971], "Proper Bayes Minimax E s t i m a t o r s o f the M u l t i v a r i a t e Normal Mean," Annals o f Mathematical S t a t i s t i c s , 42, 385-388.  32.  Strawderman, W. [1973], "Proper Bayes Minimax E s t i m a t o r s o f the M u l t i v a r i a t e Normal Mean V e c t o r f o r the Case of Common Unknown V a r i a n c e s , " Annals o f S t a t i s t i c s , 1, 1189-1194.  MacMillan'.  

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