UBC Theses and Dissertations

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UBC Theses and Dissertations

The ratio, mean-of-the ratios and Horvitz-Thompson estimators under the continuous variable model Chamwali, Anthony Alifa 1974

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r  THE RATIO, MEAN-OF-TH E-RATIOS  I  AND  HORVITZ-THOMPSON ESTIMATORS UNDER THE CONTINUOUS V A R I A B L E MODEL BY ANTHONY A L I F A CHAMWALI B. SC., UNIVERSITY OF EAST A F R I C A THE U N I V E R S I T Y COLLEGE, DAR-ES-SALAAM, 1 9 7 0  A THESIS SUBMITTED  IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (BUS. ADMIN.) IN THE  FACULTY OF  COMMERCE AND  V/E ACCEPT  BUSINESS  ADMINISTRATION  THIS THESIS AS CONFORMING TO  THE  REQUIRED STANDARD  THE UNIVERSITY OF B R I T I S H COLUMBIA APRIL,  1974  In p r e s e n t i n g an  this  thesis  in partial  advanced degree a t t h e U n i v e r s i t y  the  Library  I further for  shall  make i t f r e e l y  f u l f i l m e n t of the requirements f o r of British  available  Columbia,  I agree  that  f o r r e f e r e n c e and s t u d y .  agree that p e r m i s s i o n f o r extensive  copying of t h i s  thesis  s c h o l a r l y p u r p o s e s may b e g r a n t e d by t h e h e a d o f my D e p a r t m e n t o r  by  h i s representatives.  of  this  written  I t i s understood that  thesis f o rfinancial  gain  shall  n o t be a l l o w e d w i t h o u t my  permission.  Department o f Commerce and B u s i n e s s The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Date  copying or publication  April,  1974  Columbia  Administration  ABSTRACT  This  study  i n v e s t i g a t e s the performances of  the  ratio  and  the Horvitz-Thompson  continuous  estimator,  the mean-of-the-ratios (HT) e s t i m a t o r  v a r i a b l e model Under t h i s  is  t o the i n v e s t i g a t o r  related  t o an a u x i l i a r y  Y(Xj_) = e{X £(Zi|  Xi)  2  ±  =  £ (ZiZj / XiXj)  It  i s assumed,  over  i n this  (0, CP) w i t h  estimated, design  variable,  where  cr (Xi)  x?  2  ( i* j)  p a p e r , t h a t X i s gamma r.  the a d d i t i o n a l  f u n c t i o n , P(x),  e (0, ^ )  = k  =0  parameter  under  X, by  VXi 2  is  which  i s assumed t o be  )  x^) = 0  £(z \  (1972a,  m o d e l , t h e c h a r a c t e r , Y,  + Z(Xi)  ±  under t h e  o f C a s s e l and S a r n d a l  1972b, 1 9 7 3 ) . of i n t e r e s t  estimator  distributed  The mean o f Y i s t o be assumptions  l ) polynominal  that the 2) e x p o n e n t i a l ,  i.e.  It  1)  P(X)  -  2)  P(X)  =  i s observed  that  f o r a wider  ratios  estimator  r  e  c  x  f o r g = 0 o r 1, t h e r a t i o  performs b e t t e r than and  (1-c)  range  the other of v a l u e s  two.  estimator  F o r g = 0,  1 o r 2,  o f rn o r c , t h e mean-of-the-  performs b e t t e r than  t h e HT  estimator.  When P ( X ) i s p o l y n o m i n a l , t h e III e s t i m a t o r i s most efficient  i f the sampling  design  i s approximately  The r e s u l t s c o m p a r e w e l l w i t h t h o s e under  similar  assumptions.,  ii  of o t h e r  pps.  researchers  . TABLE OF CONTENTS Page ABSTRACT  i  ACKNOIVLEDG EMENT S 1  2  3  4  .  '  '.  v i  INTRODUCTION C l a s s i c a l Sampling Theory. . The S a m p l e S u r v e y T h e o r y . . . . . . . . S e a r c h f o r O p t i m a l E s t i m a t o r s and Sampling Designs. . . The R a t i o , M e a n - o f - t h e - r a t i o s a n d HT E s t i m a t o r s .  1 1 4  10  THE PROBLEM Statement of t h e Problem . . . . . . . . M e t h o d U s e d and P r o b l e m s o f E v a l u a t i o n .  14 14 17  THE RESULTS Results Results Results Results Summary  21 21 23 37 45 52  f o r P ( x ) P o l y n o m i n a l a n d k = 0.5 f o r P ( x ) P o l y n o m i n a l and k = 1.0' f o r P \ x ) E x p o n e n t i a l a n d k = 0.5 f o r P ( x ) E x p o n e n t i a l a n d k = 1.0 o f R e s u l t s and Some C o n c l u s i o n s  DISCUSSION OF RESULTS Some E m p i r i c a l C o m p a r i s o n s . . . . . . . Comments and Some I m p l i c a t i o n s . . . . . Some L i m i t a t i o n s Some R e c o m m e n d a t i o n s  REFERENCES  .'  iii  6  .55 55 60 64 65 66  L I S T OF TABLES Table Number I  II  Pac^e  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s P o l y n o m i n a l , k = 0.5, g= 0 V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d P ( x ) i s P o l y n o m i n a l , k = 0.5, g =1 A  III  • 0 / 1 2 4  ^  :  l  o  ^  V a r i a n c e s o f M-YHT a n d Y- Ymr when X i s Gamma D i s t r i b u t e d , P ( x ) i s P o l y n o m i n a l , k = 0„5,  g=  IV  2 V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s P o l y n o m i n a l , k = 1,  2  g =o  V  3 0  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s P o l y n o m i n a l , k = 1,  g  = i ,  VII  VIII  V a r ( N Y H T a n d V a r (K-Ymr) when X i s Gamma Distributed P(x) i s Polynominal, k - 1 , g =2 V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 0.5, g =0  55  i  XI XII  4  3<  4  _ ?  1  V a r ( M Y H T ) a n d V a r ( M Ymr) when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 0.5,  g =2  X  3  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x o o n e n t i a l , k = 0.5,  g  I IX  2  A  A  VI  3  4  3  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 1, g = 0 V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 1, q = 1 Var( KYHT) a n d V a r ( K Y m r ) when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 1, o = 2 iv  A6 4  8  5  0  8  L I S T CF  FIGURES  Figure Number 1  2  3  4  5  Page V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s P o l y n o m i n a l , k = 0.5, g = 0 .  2o  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s P o l y n o m i n a l , k = 0.5, g = 1 .  27  V a r i a n c e s o f KYHT and M-Ymr when X i s Gamma D i s t r i b u t e d , P ( x ) i s P o l y n o m i n a l , k = 0.5, g = 2  29  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s P o l y n o m i n a l , k = 1, g = 0  31  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s P o l y n o m i n a l , k = 1, g = 1 A  6  7  8  A  9  10  11  12  33  V a r (HYHT) and V a r (KYmr) when X i s Gamma D i s t r i b u t e d , P ( x ) I s P o l y n o m i n a l , k = 1, 9 = 2  35  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 0.5, g = 0  40  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k - 0.5, g = 1 . . .  42  A  A  I  '  V a r (MYHT) a n d V a r (nYmr) when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 0.5, g = 2  44  V a r i a n c e s o f . t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 1, g = 0 . .  47  V a r i a n c e s o f t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 1, g = 1  49  A (MYHT) and V a r  Var Distributed, 0 = 2  A  (f^Ymr) when X i s Gamma P ( x ) i s E x p o n e n t i a l , k = 1, ".  v  51  ACKNOWLEDGEMENTS for  I would l i k e  t o thank  D r . C. E. S a r n d a l  h i s i n v a l u a b l e a d v i c e and h e l p o f f e r e d  t h i s study.  I would a l s o l i k e  t o extend  the C a n a d i a n Government f o r s p o n s o r i n g here  i n Canada.  Tanzanian of  Lastly,  Government and t h e P r i n c i p a l  arranged  f o r my  my' t h a n k s t o education  my a p p r e c i a t i o n t o t h e  D e v e l o p m e n t Management, T a n z a n i a ,  another,  my  throughout  study.  vi  of t h e I n s t i t u t e  who, i n one way o r  CHAPTER I INTRODUCTION Recently, much emphasis has been placed on the d i s t i n c t i o n between General S t a t i s t i c a l Theory and Sample Survey Theory.  I t has been f e l t that the two  t h e o r i e s should e i t h e r be r e c o n c i l e d or be t r e a t e d differently. C l a s s i c a l Sampling Theory In General S t a t i s t i c a l Theory, Sampling Theory i n v o l v e s making inferences from observed sample values t o an i n f i n i t e h y p o t h e t i c a l p o p u l a t i o n .  The sample  values are assumed t o have been drawn randomly from a smooth d e n s i t y f u n c t i o n defined on the u n i t s of the hypothetical population.  Godambe (1969, 1970) suggests  that the tendency t o a s s o c i a t e t h e o r e t i c a l s t a t i s t i c a l sampling w i t h an i n f i n i t e h y p o t h e t i c a l population  that  has a smooth d e n s i t y f u n c t i o n may l i k e l y be due t o the V|ay General S t a t i s t i c a l Theory developed. statisticians  The e a r l y  were mainly i n t e r e s t e d i n b i o l o g i c a l and  s o c i o l o g i c a l phenomena l i k e i n h e r i t a n c e .  They assumed  that some chance mechanism operated behind these phenomena, and that the chance mechanism uniquely determined the frequency f u n c t i o n of the c h a r a c t e r i s t i c under study i n the h y p o t h e t i c a l  population.  2.  When d i s c u s s i n g Sample Survey Theory, some o p t i m a l i t y p r o p e r t i e s of e s t i m a t o r s from G e n e r a l Statistical  Theory a r e sometimes encountered.  The  f o l l o w i n g a r e some of these: Y^, Y 2 , .  Let  .  ,iY  be a sample from a  n  population with density f unctionf(Y,0). ©  = d(Yi, Y , . . .  Y  2  n  be the e s t i m a t o r of the parameter 0. f u n c t i o n of t h e observed variable.  ) Since 6 i s a  sample v a l u e s , i t i s a random  Let  6* by any other The  Let  = d* (Yi , .. .  Y  n  )  s t a t i s t i c which i s not a f u n c t i o n of 0.  e s t i m a t o r 0 i s s u f f i c i e n t f o r 0 i f , f o r each 0*, the  c o n d i t i o n a l d e n s i t y of 0* g i v e n 0, P(0*J0) does not c o n t a i n 0.  A sufficient  statistic,  i f i t exists,  contains  a l l t h e i n f o r m a t i o n about the parameter t o be e s t i m a t e d . The  e s t i m a t o r , 8, I s an unbiased A  if  i t s expected  v a l u e , E ( 6 ) , e q u a l s 0.  Minimum V a r i a n c e Unbiased an unbiased  e s t i m a t o r of 0 A  Estimator  0 i s a Uniformly  (UMVUE) of 0, i f 0 i s  e s t i m a t o r of 0 and i t s v a r i a n c e i s l e s s than or  equal t o the v a r i a n c e of any other p o s s i b l e e s t i m a t o r , 6p,  of 9  i . e . Var (e)^Var (0p) f o r a l l 0p. Let 0n = dn ( Y i , . . , Y ) c  n  be an e s t i m a t o r of 9 based on a sa_mple of s i z e n.  The  3  «  sequence [0 ^ i s a c o n s i s t e n t e s t i m a t o r of 0 i f , f o r every ;r>o n  l i m p(e -£ ^e <e +e) = i , ve n  0  i.e , B  n -*cQ approaches 0 as n g e t s l a r g e .  n  The  sequence  [®n^ ^  s a  Best A s y m p t o t i c a l l y  Normal (BAN) E s t i m a t o r of 0 i f A  (a)  The d i s t r i b u t i o n  (0  ^  of  - 0)  n  approaches the  normal d i s t r i b u t i o n w i t h mean 0 and v a r i a n c e cf  2  (e) as n approaches  infinity  F o r every £ > 0,  (b)  l i m P [j ©  n  - ej>i] = o ,  ve  n (c)  There i s no other  sequence of c o n s i s t e n t  e s t i m a t o r s 0j[ , 0"! , distribution  ....,0^  . ... f o r which the  - 9)  approaches the normal  of ^ ( 6 *  d i s t r i b u t i o n w i t h mean 0 and v a r i a n c e  0*2 such t h a t  (e)  and  ^Jel_> I 0 * 2  (0)  f o r a l l 0 i n some open i n t e r v a l . I f the unbiased  0 = d(Yi, Y  ..Y ) i s a l i n e a r 2> ' n f u n c t i o n of Y and i f Var ( 0 ) i V a r (0p) f o r a l l l i n e a r 0p, A  1  A.  then 0 i s a U n i f o r m l y Best L i n e a r Unbiased  Estimator  (UBLUE) of 0. I t may be worthwhile n o t i n g t h a t w h i l e  general  s t a t i s t i c a l t h e o r y aims a t making i n f e r e n c e s about some frequency  f u n c t i o n of a h y p o t h e t i c a l p o p u l a t i o n , i t may  4.  actually  be i n f e r r i n g  (discussed  above) t h a t  t i o n s w h i c h may have population The  about  the possible  produced  nothing  the given  t o do w i t h  Sample S u r v e y  finite.  i n the General  sampling  Statistical  like  sampling  This  dealt  i s the  w i t h by t h e  sense t h e problems of theory  may be  different. survey  sampling,  and f i n i t e design  Occasionally, nonprobability the  In t h i s  t h e ones  Theory.  i n t h e two m o d e l s o f s a m p l i n g  t a k e n t o be  with real  theories.  considered  o f e l e m e n t s w h i c h a r e r e a l and  main d i f f e r e n c e b e t w e e n t h e p o p u l a t i o n s  In  the hypothetical  the populations  They a r e not h y p o t h e t i c a l  considered  inference  s e t of o b s e r v a -  Theory  They c o n s i s t  countable.  two  mechanism  at a l l .  I n Sample S u r v e y T h e o r y , are  chance  following  since  populations,  and t h e e s t i m a t o r  circumstances sampling  the investigator he h a s c h o i c e  over t h e  he may want t o u s e .  may n e c e s s i t a t e  altogether.  situations that  deals  lead  Cochran  t h e use of (1963)  gives  to nonprobability  sampling: (1)  When t h e sample population that  i s r e s t r i c t e d t o a part i s readily accessible.  of the  (2)  When t h e sample  i s selected  (3)  With a small but heterogenous p o p u l a t i o n , the sampler i n s p e c t s t h e whole of i t and s e l e c t s a s m a l l sample o f ' t y p i c a l ' u n i t s , i . e . , u n i t s t h a t a r e c l o s e t o h i s impression of the average of t h e p o p u l a t i o n . T h i s method i s s o m e t i m e s c a l l e d judgement or p u r p o s i v e s e l e c t i o n .  haphazardly.  5.  (4)  L a s t l y , when the sample c o n s i s t s e s s e n t i a l l y of volunteers.  He notes t h a t these methods may give good r e s u l t s under the r i g h t c o n d i t i o n s a l t h o u g h they are not amenable t o the development of a sampling  t h e o r y owing t o l a c k of .random  selection. But whether one d e a l s w i t h Sample Survey or G e n e r a l S t a t i s t i c a l Theory, same problem  Theory  he i s d e a l i n g w i t h the  of i n f e r r i n g from the sample t o the p o p u l a t i o n  u s u a l l y w i t h the h e l p of P r o b a b i l i t y Theory In the G e n e r a l S t a t i s t i c a l Theory, n e a r l y a l l the sampling  and S t a t i s t i c s .  random sampling  d e s i g n problems.  With  p o p u l a t i o n s , t h i n g s are not t h a t much easy. the p o p u l a t i o n may make random sampling  solved  survey  The nature of  difficult;  i t may  not be p o s s i b l e t o i d e n t i f y a l l the u n i t s i n the p o p u l a t i o n . In most c a s e s , the frequency f u n c t i o n of the c h a r a c t e r under c o n s i d e r a t i o n i s unknown.  T h i s makes i t d i f f i c u l t  t o check some of the o p t i m a l i t y c o n d i t i o n s of e s t i m a t o r s i n the G e n e r a l S t a t i s t i c a l Theory.  And o p t i m a l e s t i m a t o r s  under the G e n e r a l S t a t i s t i c a l Theory the Sample Survey Theory. of  finding  sampling  need not be so under  T h i s r a i s e s the main  problem  d e s i g n s and e s t i m a t o r s t h a t a r e  o p t i m a l f o r the sample survey s i t u a t i o n .  Mathematically,  i n the sample survey model, we have a set P of N u n i t s t h a t c o n s t i t u t e the p o p u l a t i o n , P = (Ui,, U , 2  UN)  6. where  stands f o r u n i t i . An unknown q u a n t i t y Y i  which i s of i n t e r e s t t o the surveyor i s a s s o c i a t e d w i t h U^.  The surveyor wants t o know  0 = ©(Yi, Y ,  Yj«j)  2  He may a l s o know the a u x i l i a r y v a r i a b l e X , a = (Xj_, X , 2  a s s o c i a t e d w i t h P. • !  . . . ,  s = (u  Y = (Y  u  l f  2  l v  ,  u  He then  Y ,  Y )  2  n  The sampling  I }  Y, 2  Y ) n  The problem i s how t o s e l e c t S  t o a s s i g n the p r o b a b i l i t y t h a t u n i t  included  plan  calculates  9 = e(Y as an e s t i m a t o r of 0.  )  n  the c o r r e s p o n d i n g  (response e r r o r s a r e i g n o r e d ) .  (how  XN)  He s e l e c t s a subset  Of u n i t s of P and observes  generates S.  i.e.  i of P w i l l be  i n S) and how t o choose the random v a r i a b l e 0  t o get an e s t i m a t e which i s as near 0 as p o s s i b l e . Certainly,  i t i s not j u s t a matter  'representative' I  sample and c a l c u l a t i n g  and Median of the sample v a l u e s . representative  of g e t t i n g a the mean, v a r i a n c e  B e s i d e s , what i s a  sample?  Search f o r O p t i m a l  E s t i m a t o r s and Sampling  Designs,  In 1934 Neyman i n t r o d u c e d t h e Gauss-Markov theorem t o o b t a i n a l i n e a r unbiased  minimum v a r i a n c e estimate f o r  7.  the mean of a survey p o p u l a t i o n . for  He e s t a b l i s h e d  that,  simple random sampling, the sample mean was the  minimum v a r i a n c e l i n e a r unbiased estimate of the survey p o p u l a t i o n mean. sampling  T h i s was an attempt  t o f i t survey  i n t o the h y p o t h e t i c a l p o p u l a t i o n model, and h i s  f i n d i n g s s t i m u l a t e d most sample-survey  s t a t i s t i c i a n s to  f i n d , w i t h the h e l p of Gauss-Markov theorem,  efficient  ( i . e . minimum v a r i a n c e ) unbiased e s t i m a t o r s f o r a v a r i e t y of more complex d e s i g n s .  Sampling  procedures  sampling w i t h a r b i t r a r y p r o b a b i l i t i e s , cluster  sampling, 2-stage  like,  stratified  sampling,  sampling, m u l t i - s t a g e sampling,  e t c . were designed t o reduce the v a r i a n c e of the e s t i m a t o r s (see, f o r example, Goodman and K i s h (1950), D u r b i n (1953), H a r t l e y and Rao (1962), Cochran H o r v i t z and Thompson (1952) attempted a g e n e r a l method f o r d e a l i n g w i t h sampling replacement  from a f i n i t e  (1963). t o provide  without  p o p u l a t i o n when v a r i a b l e  p r o b a b i l i t i e s of s e l e c t i o n a r e used t o the elements remaining p r i o r t o each draw.  They, i n p a r t i c u l a r ,  c o n s i d e r e d a g e n e r a l e s t i m a t o r of the p o p u l a t i o n t o t a l of the  form  -A  n Y = Iri Yi i=l where  ^± ( i = 1,  weight  f o r the i  t  N) i s a c o n s t a n t t o be used as a n  u n i t whenever i t i s s e l e c t e d f o r the  8. sample.  L e t t i n g P(Xj_) to be  element w i l l be  p r o b a b i l i t y t h a t the i " ^  i n c l u d e d i n a sample of s i z e  n,  they  showed t h a t  P(Xi) makes Y unbiased value  and  (X^  of minimum v a r i a n c e  i s the  of the a u x i l i a r y v a r i a b l e a s s o c i a t e d w i t h u n i t i ) . Godambe (1955, 1965)  considered  the  general  estimator n -21 b s i Y i i=l  •A  Y  =  where b s i i s d e f i n e d i n advance f o r a l l the N p o s s i b l e S (samples),  and  f o r a l l i i n S.  more g e n e r a l c l a s s of e s t i m a t o r s than the by H o r v i t z and  Thompson.  He  n  logically  This.is a ones c o n s i d e r e d  proved the non-existence  a u n i f o r m l y minimum v a r i a n c e  (UMV)  unbiased  of  estimator i n  t h i s c l a s s . o f e s t i m a t o r s f o r any d e s i g n P ( s ) , e x c e p t i n g those  i n which no two  common and  one  S w i t h P(S) > o have at l e a s t  uncommon u n i t .  Because of the non-existence estimator,  criterion  of UMV  other c r i t e r i a of goodness of an  were sought.  of a d m i s s i b i l i t y  p-unbiased d e s i g n P(S)  unbiased estimator  Godambe and J o s h i (1965) c o n s i d e r e d  Thompson (HT)  one  estimator  proved t h a t the H o r v i t z -  i s admissible  e s t i m a t o r s of Y , such t h a t  and  the  i n the c l a s s of a l l  the p o p u l a t i o n t o t a l ,  for  any  9.  2  TTj =  S : )  The  p ( ) > o , vuj ' s  uj  admissibility criterion,  however,  s a t i s f i e d by many other e s t i m a t o r s , and criteria (Basu, these  like  1971)  other  were i n t r o d u c e d .  e s t i m a t o r , YHT,  some q u e s t i o n s  For example, the H o r v i t z and i s uniquely  regarding  whatever  sample  lead to d i s a s t r o u s r e s u l t s  of  Thompson  'hyper-admissible'  the c h a r a c t e r under i n v e s t i g a t i o n or the may  bestness'  A d e t a i l e d examination  optimality properties raise  the YHT  new  ' h y p e r - a d m i s s i ' b i l i t y , ' 'necessary  t h e i r relevance.  But  was  design.  (Sarndal, 1972;  Basu, 1971). Many v a r i a t i o n s of the r e g r e s s i o n type  population  model y  i  = a x i + Zj_  i =  (l,...,N)  have been used, where y^ i s the c h a r a c t e r of i n t e r e s t Xj_ i s an a u x i l i a r y v a r i a b l e f o r u n i t i Zi  i s an e r r o r component.  i n the  and  population;  Cochran (1946), Godambe (1955,  1965), R o y a l l (1970, 1971)  and  many o t h e r s used the  super-  p o p u l a t i o n approach i n c o n j u n c t i o n w i t h the r e g r e s s i o n model t o compare e f f i c i e n c i e s between sampling C a s s e l and S a r n d a l  (1972a, 1972b, 1973)  v a r i a b l e model of the  (z(x)  the  continuous  form  Y(X)  where  use  methods.  =  0  (X + Z(X)  / X) = 0 , £(z(x) | X )  £(z(Xj.) z(x ) 2  | x x ) 2  2  2  =  •= o  )  cr  VXC(O.cP)  (X)  2  (x  2  *  x) 2  10. and  the a u x i l i a r y v a r i a b l e X i s assumed t o have a known  d i s t r i b u t i o n d e s c r i b e d by the d i s t r i b u t i o n f u n c t i o n F(x) (0,CP).  over  Different  n o t i o n s of unbiasedness have been used  during t h i s search f o r optimal An  e s t i m a t o r s and  e s t i m a t o r , Y f o r Y,  or p-unbiased  i s called  designs:  design-unbiased  if  E  ' t  V-  (Y) =2_  p  -  P(s)  Y  -  =  Y  stS for  a l l v e c t o r s (Yj_, .... ,Y^) , where P(s) i s a  f u n c t i o n d e f i n e d on the  set  S of subsets  probability  of s of  labels  (I,«...,N).  An if  estimator  i t i s unbiased  i s called  model-unbiased  under the assumptions of the  or£-unbiased specified  model. A  L a s t l y , an e s t i m a t o r Y i s c a l l e d  Ep^-unbiased f o r  Y if E p £ (Y) an e s t i m a t o r  can be  =IP(S) s£S  p-unbiased  C  but  (Y)  =  Y  not£-unbiased and v i c e  versa. The  PLatio,  It in  sampling  accuracy  Mean-of-the-Ratios and i s a f a c t that using designs  and  the HT  Estimators  supplementary  information  e s t i m a t o r s g r e a t l y improves  of the e s t i m a t e s .  the  Three e s t i m a t o r s t h a t have  11. r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i n the l i t e r a t u r e sample survey  t h e o r y and which make use  of  i n f o r m a t i o n are the R a t i o e s t i m a t o r , the e s t i m a t o r and  the Horvitz-Thompson  i s an a u x i l i a r y  variable,  1=1  X =  1  (Hi) e s t i m a t o r .  n I  = £x  -  i  X = NJ  .  1=1  X = — ZZ X-j, n i s the sample s i z e n i=i size. The  Mean-of-the-ratios  mean-of-the-ratios  (1.1)  X •  ,  N  I  1=1 and  X.,  Y  1  - = I n  N i s the  A  Z_  1=1  n  / x .-^-2)  y  r a t i o e s t i m a t o r s make use  of the r a t i o s  order t o improve e s t i m a t i o n sampling Ry  from an i n f i n i t e  c o n d i t i o n s are  Theorem (1)  —i- . Xj_ i n  random  p o p u l a t i o n , the r a t i o e s t i m a t o r  satisfied  (Cochran  unbiased , 1963,  The  relation  The  the  between yj_ and  X^  p.  166,  i s a straight  line  origin  v a r i a n c e of y^'about t h i s  line  i s proportional  to X i . When the v a r i a n c e of y i about t h i s 2 p r o p o r t i o n a l to X'^,  line i s  using the m e a n - o f - t h e - r a t i o s  g i v e s much b e t t e r performance than the  other  of  estimate i f  6.4):  through (2)  y•  Y and X  •With simple  has been shown t o be the best l i n e a r  two  ±  estimator i s  L  The  Y,  population  v i KYmr = - H ~ n i=i X A  If X  by  AYR = Y . X x where y =  supplementary  the r a t i o e s t i m a t o r f o r the  p o p u l a t i o n mean,KY, i s g i v e n  n I  of  estimator  estimator.  12. When the. r e l a t i o n between y± and Xj i s l i n e a r but the l i n e does not go through the o r i g i n , the r e g r e s s i o n estimator  . y"l  =  r  + b (X - x)  y  performs much b e t t e r than the otherj- Y j [ reduces t o y i f x  y  A  b = 0 and toKRY i f b = estimators but w i t h  Both the r a t i o and r e g r e s s i o n  a r e c o n s i s t e n t and, g e n e r a l l y ,  sampling d e s i g n s l i k e  b i a s of the r a t i o e s t i m a t o r has l e d t o s e a r c h i n g estimators. search. other  slightly  stratified  biased,  sampling, the  may be c o n s i d e r a b l e .  This  unbiased or b e t t e r r a t i o - t y p e  J . N . K . Rao (1969) g i v e s some r e s u l t s of the  The r e s u l t s i n d i c a t e t h a t under c e r t a i n c o n d i t i o n s ,  ratio-type estimators  perform b e t t e r than the t r a d i -  . t i o n a l r a t i o estimator. The Horvitz-Thornpson e s t i m a t o r  KYHT =  i s g i v e n by  .  ni=lP(xi)  (1.3).  where P ( x i ) i s the p r o b a b i l i t y t h a t a u n i t with v a r i a b l e x^ w i l l be i n c l u d e d of the HT e s t i m a t o r  i n the sample.  may be n e g a t i v e ,  •  auxiliary  The v a r i a n c e  or i t may not.reduce  t o z e r o even when a l l the Y-values are the same and the variance  should  a c t u a l l y be z e r o .  In many s t u d i e s the HT e s t i m a t o r  has been shown t o  compete very -well w i t h the r a t i o e s t i m a t o r s . linear  Assuming a  s t o c h a s t i c model of the form Y  i  = cc + p X i + Z  Foreman and Brewer  i  ,  =  0%  2Y t  (1971) showed that the HT e s t i m a t o r i s  13. more e f f i c i e n t tion  ratio  fraction  estimator  estimator.  Rao  ,  estimator  but they  i s l a r g e and  may  be  estimator  and t h e HT  the r a t i o  estimator  (Sukhatme  than  seleci f the  the the HI.  some c o n d i t i o n s u n d e r  which  i s superior to both  estimator.  estimator  equal  caution that  made more e f f i c i e n t  (1967) g i v e s  mean-of-the-ratios  with  a i s appreciable,  rnean-of-the-ratios estimator  ratio  !  the r a t i o i f Y?^  probabilities  sampling  the  than  Unfortunately,  i s not c o n s i s t e n t l i k e , and Sukhatme,  1970,  the the say,  p. 1 6 0 ) .  14. CHAPTER I I THE  PROBLEM  Statement of the Problem The it  problem w i t h simple random sampling  does not take i n t o account  the p o s s i b l e  of the l a r g e r u n i t s i n the p o p u l a t i o n . sampling  designs l i k e  proportional to size  sampling w i t h  importance  Because of t h i s ,  probability  (pps), and g e n e r a l l y known as  sampling w i t h v a r y i n g p r o b a b i l i t i e s came t o be These are more complex d e s i g n s t h a n sampling.  I t was  whether sampling  i s that  also realized  used.  simple random  t h a t i t makes a d i f f e r e n c e  i s done w i t h replacement  or without  replacement.  However, the e s t i m a t o r s c o n s i d e r e d were v e r y  complicated.  To get simple e s t i m a t o r s , sampling  l i k e the Midzumo system  of sampling,  of sampling, S y s t e m a t i c Sampling e t c . were i n t r o d u c e d . t h a t performances  the N a r a i n Method  with varying p r o b a b i l i t i e s  I t i s v e r y c l e a r , from  such  of e s t i m a t o r s can be improved  only making use of supplementary choosing the proper  sampling  schemes  by,  i n f o r m a t i o n , but a l s o by  mean-of-the-ratios  e s t i m a t o r and the Horvitz-Thompson e s t i m a t o r have  survey l i t e r a t u r e .  not  design.  The r a t i o e s t i m a t o r , the  quite e f f i c i e n t l y  studies,  performed  i n some of t h e . r e s e a r c h work i n sample But which of these t h r e e d-oes b e t t e r  15.  estimation design  than  used,  the  been used,.the 'bias  or  1  example, (i)  the  o t h e r s d e p e n d s on  way  class  'unbiased'  the of  the  supplementary estimators  i s defined.  sampling  information  used  and  Joshi  the  way  (1971),  for  has  states that The  HT  class  estimate  i s always  of a l l u n b i a s e d  admissible  estimates,  i n the  linear  and  non-linear. (ii)  In the  entire  estimator is (iii)  of  estimates,  i s admissible  of f i x e d  I f the  class  loss  numerical  sample  i f the  d i f f e r e n c e between the  values,  satisfies  (a)  V(t)  (b)  f o r every  is  The regression results. these  type  type  V(t)  as  exp  like  continuous  and  CO,  in  ^ l ,  (-  % -  ) dt  C <*>  more g e n e r a l l y t h e  an  estimate  three  models has  estimators  estimator  and  2  of- t h e s e  I would  three  ratios  use  estimated  2  mean, and  always a d m i s s i b l e  i s the  K > 0  0 sample  design  only  i s non-decreasing  5 the  sampling  f u n c t i o n , V ( t ) , where t  true  HT  size,  °o  then  the  of t h e  shown t o g i v e v e r y  the  HT  ratio  v a r i a b l e Model  under  of C a s s e l  mean.  the promising  performances  estimator,  estimator)  estimate  population  estimators with  t o i n v e s t i g a t e the (the  ratio  of  mean-of-thethe  regression  arid S a r n d a l  (1973).  16.  I would l i k e  t o c o n s i d e r the model  Y ( x i ) = 6(xi) + z ( x i ) ) £(z/i| X i ) = 0  V x i € (0, *P)  . £ ( 2 | x i ) = cr (xi) (  Z i  XJX-J)  zj j  =0  where Y i i s t h e v a l u e  k  2  Xi9  ( i T j)  of t h e c h a r a c t e r under  f o r u n i t i , x i i s the v a l u e variable.  =  2  Z i  E  where  investigation  of the c o r r e s p o n d i n g  auxiliary  I f u r t h e r assume t h a t the a u x i l i a r y v a r i a b l e X  i s gamma d i s t r i b u t e d  over (O, ^ ) i . e . 0  . f(x) =  X ~ r  1 e  '"  1  X  for x £ ( 0 , < ^ )  Hence i t s mean E(x) = §  x f(x)dx = r  0  My aim i s t o estimate  the p o p u l a t i o n mean  E €(Y(x) ) = P  The  =Ky  t h r e e e s t i m a t o r s of t h e p o p u l a t i o n mean t h a t I would  l i k e t o consider are *  n  MYHT = i YI  1=1  ^^mr  - £  y(*i)  -r-T-  ( 2 , I )  (2.2)  1=1  A  n  H R = Y  r  i f !  n  i=l  ( .3) 2  *i  17.  Where the d e s i g n f u n c t i o n , probability xi,  P(x^) i s the s e l e c t i o n  d e n s i t y of a u n i t w i t h a u x i l i a r y  n i s the number of u n i t s  made independently distribution  i n the sample.  variable Draws are  of each other a c c o r d i n g t o the same  of i n c l u s i o n  probabilities.  H rvitz-Thompson estimator, 0  ( 2 . l ) i s the  (2.2) i s the rnean-of-the-  r a t i o s e s t i m a t o r and (2.3) i s the c l a s s i c a l r a t i o The  estimator.  t h r e e e s t i m a t o r s are Ep£-unbiased and t h e i r  MSE's, then,  equal t h e i r  EpGvariances.  Method used and Problems of E v a l u a t i o n s The evaluated obtained  e f f i c i e n c i e s of these  e s t i m a t o r s w i l l be  i n terms of t h e i r EpGvariances,  which a r e  from Var  (KY)  =  E £ ( f t ) - (E £(fiy) ) 2  p  y  2  p  G e n e r a l l y , the e x p r e s s i o n s f o r the v a r i a n c e s of these estimators are x__±_k 2xi f ( x ) d ( x )  Var (RYHT)  Var  Var  2  A (Hymr)  Ky n  x  °o  j k x ~ 2  g  2  p(x) f ( x ) dx  0  A  (KYR)  p(x)  where P ( ) = J T P(x,) -' i=l -  1  ,  - 1)  f(x) = U f ^ ) I - I  x  ,  f ( x j d(x_)  d(x) = J J d x l - l  i  18. and  the l a s t  integral  f o r i = 1,  i s n-dimensional  (over  0 z_ x ^ , / ^  n ) . The v a r i a n c e s depend e s s e n t i a l l y on  t h r e e t h i n g s , namely, the shape o f the c o n t i n u o u s distribution,  f ( x ) , of the a u x i l i a r y v a r i a b l e ,  v a r i a n c e of Z i and the s e l e c t i o n I will 2)  probability  compare the v a r i a n c e s under  x; the  density, P ( x ) .  l ) polynomial p(x),  exponential p(x).  (1)  Let )  -  P(x  ?M  s  (m = 0 r e p r e s e n t s pps sampling  X  m  simple  scheme).  Var  ^  sampling,  estimators are:  [ f r { f T ^ (JU+r-mH  ^jg-r+a^).  (tfymr) =  m = 1 represents  In t h i s case the e x p r e s s i o n s f o r  the v a r i a n c e s of these Var rf(YHT) =  random  k  2  P(g+r-mjj-l}(2 4) 0  -By .  (  2  2 o 5  )  f o r g = o: k  A V a r  (  H  Y  R  )  2  2  (n^nr-l)(nm-nr-2) ' H y  =  —-  '  Uy^  k2  ftrTTT ' 2  ( i f  n  i  s  l a  ?ge)  .  (2.6)  "  f o r g = 1: Var  (KYR) = -n-  ^ • U_ nm+nr-1 / iiyi n  • k -. m+r 2  2  (for n large)  (2.7)  19. I e v a l u a t e these e x p r e s s i o n s n u m e r i c a l l y f o r r = 1, 2, 3 and  k and g f i x e d at 0.5 0.5  respectively,  0.5  and 0 r e s p e c t i v e l y ,  and 2 r e s p e c t i v e l y ,  and 1  1 and 0 r e s p e c t i v e l y ,  e t c . , and as m takes the v a l u e s -1 ( 0 . 5 )  2,3,4.  I will  observe how the v a r i a n c e s of these e s t i m a t o r s behave. (2)  Let P(x)  (c  =  (l-c)  e  r  ,  c x  c£.l '  = 0 r e p r e s e n t s simple random sampling).  In t h i s case  the e x p r e s s i o n s f o r the v a r i a n c e s a r e : Var  (KYHT)  = ^n C ^ — f !l2±X-L_ p U - c ) * l ( r ) L ( .)2+r  (  (  +  c+a  Var  fr(vrnr1 = H y .  .  g+U  2  c + 1 )  )  3  J  k V (q+r 2)  2  2  7  (2.9) for  g=0:  Var (KyR)  =  K? y  n i ^ ^ c l (nr-l)(nr-2l 2  1  f^'.  for  g=l:  Var  (kyR)  =  K  iiHllrcJ  2 (  f  o  r  n  l  a  r  g  e  )  {  2  <  1  )  k (l-c) nr-1 2  v  ^ix n  2 •  k (l-c). r 2  (  f o r  n  large)  (2.1l)  As with the polynomial case, these e x p r e s s i o n s are e v a l u a t e d and compared as v a l u e s of It  0  has. not been p o s s i b l e  r » c, g, k v a r y .  to evaluate' the v a r i a n c e  20.  of t h e r a t i o  e s t i m a t o r when g = 2, o n l y t h e v a r i a n c e s  o f t h e HT e s t i m a t o r be  and m e a n - o f - t h e - r a t i o s  c o m p a r e d when g = 2.  variance  will  With exponential p ( x ) , the  of the mean-of-the-ratios  only i f r + g>2.  estimator  estimator  i s finite  T h r o u g h o u t , I have u s e d t h e a p p r o x i m a t e  A. expressions  o f V a r (KyR)  f o r my e v a l u a t i o n s .  a r e t a b u l a t e d b e l o w and i l l u s t r a t e d  by g r a p h s .  The  results  CHAPTER 3 THE RESULTS R e s u l t s f o r P(x) Polynominal The illustrated  results  and k = 0.5  a r e presented  i n Tables  I - I I I and  by f i g u r e s 1-3.  When g = 0 the r a t i o e s t i m a t o r has, g e n e r a l l y , the s m a l l e s t v a r i a n c e f o r f i x e d v a l u e s of r and m, f o l l o w e d by the m e a n - o f - t h e - r a t i o s estimator.  e s t i m a t o r and then by the HT  When r = 1 not much comparison can be made  between the HT e s t i m a t o r and the r n e a n - o f - t h e - r a t i o s e s t i m a t o r as the v a r i a n c e of the HT e s t i m a t o r f o r -l.O^m^l w h i l e the v a r i a n c e of the estimator  i s finite  f o r m>l.  mean-of-the-ratios  In the range of v a l u e s of m  f o r which the v a r i a n c e s of t h e r a t i o e s t i m a t o r either  the HT e s t i m a t o r  i s finite  and of  or the m e a n - o f - t h e - r a t i o s  estimator  are f i n i t e , t h e r a t i o e s t i m a t o r has s m a l l e r v a r i a n c e . r=2  and 0<m<2 where the v a r i a n c e s of a l l the t h r e e  are f i n i t e ,  When  estimators  the r a t i o e s t i m a t o r has the s m a l l e s t v a r i a n c e ;  I  the HT e s t i m a t o r  i n i t i a l l y has a smaller v a r i a n c e than the  mean-of-the-ratios estimator takes  estimator.  The v a r i a n c e  i t s minimum value a t about m = 1.0 and f o r  m?1.0 the m e a n - o f - t h e - r a t i o s  estimator's variance i s  s m a l l e r than t h a t of the HT e s t i m a t o r . HT e s t i m a t o r  of the HT  starts  at i n f i n i t y ,  takes  The v a r i a n c e of the i t s minimum value a t  22. m = 0.5 when r = 1, and m = 1.0 when r = 2 o r 3, and goes to  i n f i n i t y a s rn t a k e s h i g h e r v a l u e s ,  r a t i o estimator  When r = 3 t h e  i s most e f f i c i e n t f o l l o w e d by the mean-of-  t h e - r a t i o s estimator  and the.'HT e s t i m a t o r  i s last.-  v a r i a n c e s of the r a t i o and m e a n - o f - t h e - r a t i o s get  smaller and s m a l l e r as m i n c r e a s e s .  w i t h changes i n the value observing  formulas  of r .  The  estimators  The same i s t r u e  T h i s can a l s o be seen by  (2.5) and (2.6) f o r the v a r i a n c e s of  the two e s t i m a t o r s . When g = 1 the r a t i o b e t t e r than the other two.  estimator  still  When r = 1 and m^l, the HT  e s t i m a t o r has s m a l l e r v a r i a n c e than the estimator.  performs  mean-of-the-ratios  The r e v e r s e i s t r u e f o r m^l.  When r = 2 or 3, the m e a n - o f - t h e - r a t i o s has  s m a l l e r v a r i a n c e than the HT e s t i m a t o r  v a r i a n c e of the HT e s t i m a t o r t o the v a l u e estimator.  takes  estimator  except  when the  i t s minimum v a l u e  of the v a r i a n c e of the  equal  mean-of-the-ratios  When r = 1 the HT e s t i m a t o r b e a t s the mean-of-  t h e - r a t i o s estimator  f o r l<m, the m e a n - o f - t h e - r a t i o s  e s t i m a t o r b e a t s the HT e s t i m a t o r when m?l. the HT e s t i m a t o r takes  i t s minimum value  The v a r i e n c e of  a t m = 1.0, and  the v a r i a n c e s of the r a t i o and m e a n - o f - t h e - r a t i o s  estimators  become s m a l l e r and s m a l l e r as m gets l a r g e r and l a r g e r . When g = 2 the m e a n - o f - t h e - r a t i o s v a r i a n c e i s a constant  estimator's  and smaller" than t h a t  ox the HT  23. estimator  (except when the v a r i a n c e of the HT e s t i m a t o r  takes i t s minimum value equal t o the value of the m e a n - o f - t h e - r a t i o s  estimator).  of the v a r i a n c e  The v a r i a n c e of the  HT e s t i m a t o r i s s m a l l e s t a t m = 1.0.  1  .  G e n e r a l l y , as m i n c r e a s e s , the v a r i a n c e s of the r a t i o and the m e a n - o f - t h e - r a t i o s  e s t i m a t o r s are reduced,  but the extent t o which these v a r i a n c e s can be reduced by, say, a g i v e n b i g m i s l e s s e n e d as g takes h i g h e r v a l u e s . At m = 4, f o r example, the v a r i a n c e s of the r a t i o mean-of-the-ratios  e s t i m a t o r s are approximately  and the  zero f o r  g = 0, 0.05 M, /n f o r g = 1, and. the mean=of-the-ratios y e s t i m a t o r has a constant v a r i a n c e of 0.25|{ /n when q = 2. y 2  2  For l a r g e v a l u e s of m, the d i f f e r e n c e between the v a r i a n c e of t h e r a t i o  e s t i m a t o r and t h a t of the m e a n - o f - t h e - r a t i o s  estimator i s very v a l u e of r .  small and not very much dependent on the  The s m a l l e s t p o s s i b l e v a l u e taken by the  v a r i a n c e of the HT e s t i m a t o r and the extent t o which i t i s finite  i s g r e a t l y a f f e c t e d by the v a l u e r t a k e s .  As r  takes b i g g e r v a l u e s , the v a r i a n c e of the HT e s t i m a t o r can' tlake smaller v a l u e s and i s f i n i t e  f o r a wider range of the  v a l u e s of m. R e s u l t s f o r P(x) Polynominal The  and k = 1  r e s u l t s a r e g i v e n i n t a b l e s IV-VI and  i l l u s t r a t e d by F i g u r e s 4 - 6. . When g  r  0 and r - 1, the v a r i a n c e s of the H i  TABLE I Variances P(x)  of the E s t i m a t o r s when X i s Gamma D i s t r i b u t e d ,  i s Polynominal,  k = 0.5,  g = 0.  (Each e n t r y  should  be m u l t i p l i e d byf^{ /n) 2  r  r = 1  = 2  m  -1.0 -0.5 0.0 -  0.5 1.0 1.5 2.0 3.0 4.0  A  M-YHT CO  5.28 1.25 0.57 oo oo  ^ YHT 5.13 OO 1.00 1.65 0.25 0.56 0.11 0.22 oo 0.063 0.13 0.33 0.040 0.47 0.13 0.028 C P 0.042 0.016 0.02 0.01  Kymr  HYR  •  Kymr  KYR J0.25 JO.11 i  u I UYHT  J  I 2.38 ! !  0.97 ;0.063 i 0.36 0.33 J0.040 0.11 I1 0.13 |0.028 0.042 0.067 J0.020 •0.15 0.042 |0.016 0.50 j  !  ! 1  i  0.021 j o . o i o 0.013 J0.0069 j l  = 3  r  ! i  •  A  I i  oo  .  Hymr oo  ! i >  0.33 | 0.13 I 0.067 j 0.042 j j 0.029 ! ! 0.021 | 1  KYR 0.063 0.040 0.028 0.020 0.016 0.012 0.010  0.013 ! 0.0069 ! 0.0050 0.0083 !  'TABLE I I V a r i a n c e s of the E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P(x) i s Polynominal, k = 0 . 5 , g = 1 .  (Each e n t r y should be  m u l t i p l i e d by|U. /n). 2  r  =  r  1  =  r=  2  i  3  .. !  I 1  r  m  A  KYHT  -1.0  oo  -0.5  5.48  0.0  1.25  Kymr  K R  0.5 0.25  A  HYHT  H Ymr  H-YR  oo  0.25  2.50  1.762  0.50  0.17 •  0.63  0.25  0.13  5.37 >  A  kymr.  KYHT  Y  C O  C O  A  A  A  ! !  0.25  r^YR • 0.13  1.04  0.17  0.10  |  0.42  0.13  !  j  0.083  |  i  0.37  0.5  0.17  0.25  0.25  1.5  2.52  2.0  Go  0.5 !  1.0  3.0 4.0  0.21  0.17  0.10  0.13  0.13  .0.13  0.17  0.10  0.29  0.13  0.08  0.86  0.08  0.06  oo  ;  j  i  i  0.14  0.10  0.073  |  0.083  0.08  0.083  0.063  1  0.10  0.071  0.18  0.071  0.055  j  0.08  0.063  0.50  0.063  0.050  j  0.063  0.050 I 3.17  0.050  0.043  I  j i1  0.06  0.05  i0 . 0 5 ! i  j1  0.042 j  i i  i i  1  I  cO  i 0.041 i  I  (  0.035  TABLE I I I V a r i a n c e s of YHT and Ymr when X i s Gamma D i s t r i b u t e d , P(x)  i s Polynominal,  k = 0.5, g = 2.  (Each entry  should  be m u l t i p l i e d by Ky/n)  1  ! i  ]  r = 1  i  ! I  r = 2  r = 3  A  A  J.  A  m  i  1 1 |  ;  - 1 . 0  -  0.5  KYHT  A  A  KYmr  [X, YHT  U Ymr .0.25  3.17  0.25  0.25  1.41  0.25  oo  0.25  6.50  6.36  0.25  2.21  <  KYHT  •  KYmr i  j  0.0  1.500  0.25  0.88  1  0.25  0.67  0.25  ;  !  0.5  0.47  0.25  0.38  I  0.25  0.34  0.25  .  !  1.0  0.25  0.25  0.25  1  0.25  0.25  0.25  !  1.5  0.47  0o25  0.38  !  0.25  0.34  0.25  2.0  1.50  0.25  0.86  i  0.25  0.67  0.25  3.0  0.25  6.25  I  0.25  3.17  0.25  4.0  0.25  ! !  0.25  ! 24.00 i  0.25  1 i  j  j  1  j i  • H444 • - •  .  : .  "TT  TTTT  !  EIGUHE -3; t-tr  I  !  !  '  m i M-HI  i  i ; •T  -  " I T — '  T ' T T T ?  • -- V a r i a n c a nd(KY mr) whe n • X • i b • • .' ^Gamma [ D i s t r i b u t e d p(x)'is:£diynominal,  E5TJF  (0;illustrate:fable:  [Jl 4J  1  pr  XTX  44 -h  I  jw 1  j V a i . (HYrnr; f o r  ^Vax^KYHfl 111!  !  £L  r  rr ff  H±tt±  rrr rr M 4 -4-  + -H 11! '  .rr  11  :.j  •44—h  any •  j Lu-L -Hi  Li" r  0r  •4  1414X1.-4. r  mt  ±  H  fflxllx tr U.-l-L  r r r  ttt  t  . : tfct1 XIXI  t t t  .i-i.  ;.L . lift  f or ' r • = i? :2, 3 :re pectlye! - 1 II 4441—  0'. •  f t : • -• •  • • -rt  1  4-M.1.  .14 -L. .1 —1-4. L ; i f (L  Ltt-H-  r-r r  ;xt;X -rr 4 4 t I  L.  H-hH t  1 1 I11  -4-  rrrr  txjx :!xi:  -  t-Hi  Bit  t  TABLE IV Variances P(x)  of t h e E s t i m a t o r s when X i s Gamma D i s t r i b u t e d ,  i s Polynominal,  k = 1, g = 0.  (Each e n t r y should be  m u l t i p l i e d by  r = 1  m  A  KYHT  KYmr  -1.0, -0.5  r A  jA  A  (iYR. j pi YHT 5.50  6.63  '4.00  0.0  1.00  1.00 I 0.75  0.5  1.75  0.44 j 0.40  1.0  oo  1.5  oo  I  1.33  2.0  0.50  3.00  0.17  4.00  0.083  j KYmr oo  1.08  1.32 •  0.44  0.52  0.19  0.27  0.17  0.17  i  0.36  0.12  0.27 |0.082  1.00  0.084  o°  0.040:  | 0.25  '1.33 |0.16 0.50 j0.11  j DO  0.063;  0.43  j  0.16 | 1.50 0.11  i.oo  I  j  0.25 ! 0.50  0  0  .0.17 :  Ax  | KYHT 2.50  ! 1.86  i  j  ;A  / v  | KYmr 1 f^YR  I  oo  j  = 3  •0.063  0.083J 0.04  !0.050|0.028  ;• 0.052 I 0.033  I  A  j |4YR ! 0.25 I t  I 0.16 ! 0.11 !  j  0.08 ! 0.06 j  I 0.048 ! 0.04 J |  : 0.028 i  i 0.020  TABLE V a r i a n c e s of  YHT and  Ymr when X i s Gamma D i s t r i b u t e d , P ( x ) ,  i s P o l y n o m i n a l , k = 0.5, multiplied  r  A, RYHT  m  by  j  o.o  j 0.5 j 1.0 1 1.5 | 2.0 j 1 3.0 j  j i  7.44 2.00 0.96 1.00 2.53  4.0  •  • i  OO  oo  2.00 1.00 0.67 0.50 0.33 0.25  (Each e n t r y should be  2  j  r  i  A, IKYHT HYR  -1.0 j -0.5  g = 2.  L[ /n)  = 1  fiYrnr  V  2.00 1.00 0.67 0.50 0.40 0.33 0.25 0.20  = 2  j A | KYmr.  -  = 3  1  HYR  i 6.50 1.00 i oO 2.32 j 2.00 0.67 0.50 .1.00 j 1.00 0.55 "! j 0.67 0.40 0.50 i 0.33 0.84 \i 0.50 0.40 0.29 2.00 1 0.25 ! 0.33 0.20 oO 0.25 . 0.20 0.17 I I  r  KYHT 3.00 1.36 0.67 0.37 0.33 0.50 1.00 5.67  !  j A ; j K.Ymr ;  !i  i 1.00 0.67 0.50 0.40 0.33 0.29 0.25 0.20 0.17  A KYR  ;  0.50 0.40 0.33 0.29 0.25 0.22 0.20 0.17 0.14 . . .  TABLE V I V a r i a n c e s of P(x)  YHT and  Ymr when X i s Gamma D i s t r i b u t e d ,  k = 1,  i s Polynominal  g = 2.  (Each e n t r y should  be m u l t i p l i e d by j^y/n)  r )  i  i  !  m  1— t  1  | -0.5 j \ 0.0  i I  !  1 1  j 10.78  3.00  i  0.5  ; i.o  j :  i  1.36 1.00  i  A  A,  KYmr 1  11.00  1  4.15  1  1  !  i  i  I  »  i  1  ;  1.36  i  2.0  i  3.0  1  j 3.0  i  |  '  i  4.0  1  A  KYHT  1.5 i  = 3  r  '  ; KYHT —  i -i.o  !  = 1  1  KYrnr |  A  A  KYHT  K Ymr  5.67 2.86  •  2.00  i  j  1.67  1  1.21  1  !  L17  i  !  1.00  1  1.00  1  !  1.21  1  1.17  1  2.0  1  1.67  .1  11.00  1  5.67  i  • i  i i  i  i  •  39.00  ]  ;  ! CO  36. e s t i m a t o r and of the m e a n - o f - t h e - r a t i o s  e s t i m a t o r are not  comparable as the v a l u e s of m f o r which these are f i n i t e are not the same.  variances  In the range of v a l u e s of m  f o r which e i t h e r the v a r i a n c e of the HT e s t i m a t o r or the v a r i a n c e of the m e a n - o f - t h e - r a t i o s the r a t i o e s t i m a t o r are f i n i t e , more e f f i c i e n t .  estimator  the r a t i o e s t i m a t o r i s  When g = 0, r ~ 2, and 0 < n U l  e s t i m a t o r has s m a l l e s t v a r i a n c e f o l l o w e d . b y or and then by the m e a n - o f - t h e - r a t i o s mean-of-the-ratios m">l.  and t h a t of  the r a t i o  the HT e s t i m a t -  e s t i m a t o r , but the  e s t i m a t o r beats the HT e s t i m a t o r f o r  When g = 0, r = 3 and m<0.8, the r a t i o  estimator i s  best f o l l o w e d by the HT e s t i m a t o r and the m e a n - o f - t h e - r a t i o estimator i s l a s t .  The r a t i o and m e a n - o f - t h e - r a t i o s  e s t i m a t o r s become more e f f i c i e n t w i t h an i n c r e a s e i n the value of both m and r .  The HT e s t i m a t o r ' s v a r i a n c e  takes  i t s minimum v a l u e f o r v a l u e s of m near 0.8; i t a l s o  takes  s m a l l e r v a l u e s as r i n c r e a s e s . When g - 1 the r a t i o e s t i m a t o r always has the smallest variance.  When r = 1 and m<l the HT e s t i m a t o r  i s b e t t e r than the m e a n - o f - t h e - r a t i o s  estimator.  The  ! r e v e r s e i s t r u e when m>l.  When r = 2 and f o r e i t h e r mcO  or m>l, the HI e s t i m a t o r has l a r g e r v a r i a n c e than the mean-of-the-ratios  e s t i m a t o r , f o r 0 < r n * l the HT e s t i m a t o r ' s  v a r i a n c e i s s m a l l e r than t h a t of the m e a n - o f - t h e - r a t i o s . When r = 3 the HT e s t i m a t o r b e a t s the m e a n - o f - t h e - r a t i o s f o r 0 . 3 < : r i K l and the m e a n - o f - t h e - r a t i o s the HT e s t i m a t o r  f o r other v a l u e s of m.  estimator  beats  37.  When g = 2 the v a r i a n c e of the e s t i m a t o r i s a constant  mean-of-the-ratios  at j.( /n and, except when the 2  v a r i a n c e of the HT e s t i m a t o r takes i t s minimum v a l u e , i t i s smaller than t h a t As  of the HT e s t i m a t o r .  i n the other case, an i n c r e a s e i n the value of g  has a g e n e r a l e f f e c t  of l e s s e n i n g the power of r e d u c i n g the  v a r i a n c e s of the r a t i o and r n e a n - o f - t h e - r a t i o s e s t i m a t o r s by i n c r e a s i n g m.  No improvement can be made on the v a r i a n c e  of the m e a n - o f - t h e - r a t i o s  e s t i m a t o r when g = 2.  v a r i a n c e of the HT e s t i m a t o r  The  i s s m a l l e s t f o r v a l u e s of m  near 1.0. The the e f f e c t  change i n the v a l u e  of k from 0.5 t o 1.0 has  of s l i g h t l y i n c r e a s i n g the v a l u e s of the  v a r i a n c e s of the e s t i m a t o r s f o r f i x e d v a l u e s of g, r and rn. R e s u l t s f o r P(x) E x p o n e n t i a l and k - 0.5 The  results  are g i v e n i n T a b l e s V I I - I X and  i l l u s t r a t e d by F i g u r e s 7-9. When g = 0 or 1, the r a t i o e s t i m a t o r i s most efficient  f o l l o w e d by the m e a n - o f - t h e - r a t i o s  then by the HT e s t i m a t o r . mean-of-the-ratios  e s t i m a t o r and  The v a r i a n c e s of the r a t i o and  e s t i m a t o r s are z e r o when c = 1.  The  g r e a t e r the v a l u e r takes the more e f f i c i e n t the e s t i m a t o r s become.  The graph of the v a r i a n c e of the HT e s t i m a t o r i s  u-shaped; an i n c r e a s e i n r has the e f f e c t  of pushing the  minimum p o i n t of the v a r i a n c e of the HT e s t i m a t o r to the left.  33.  When g - 2, the v a r i a n c e of the estimator of  i s constant  the HT e s t i m a t o r .  mean-of-the-ratios  at 0.25ky/n and i s s m a l l e r than The e f f e c t  of i n c r e a s i n g r on the  shape of the v a r i a n c e of the HT e s t i m a t o r ' s curve to  shift  that  i t s minimum p o i n t t o the l e f t .  i s simply  The r a t e of  improvement on the e s t i m a t o r s by i n c r e a s i n g c i s reduced g changes from 0 t o 1, and the e s t i m a t o r s become l e s s efficient too.  as  TABLE V I I Variances P(x)  of t h e E s t i m a t o r s when X i sGamma D i s t r i b u t e d ,  i s E x p o n e n t i a l , k = 0.5, g = 0.  (Each e n t r y  should  be m u l t i p l i e d by \i /^) 2  = 2  : AKYHT j KYR rr  -•  1-  i  1.00  138.58  i  0.81  18.90  j  0.64  6.00  0.49  2.50  j  0.36  1.25  |  0.25  0.70 0.52  I  |  0.16 0.09  0.60  |  0.04  1.44  i  0.01 0.00  !  r = 3  \ RYHT  KYHT !  0.25  |288.50  [  0.20  j  713.50  21.25  I  0.16  29.84  |  .4.96  0.12  4.70  j  1.61  S I  0.09  0.56  i  |  i  i  i  !  0.21  I  I  ! 1  -  -1  \  0.58 2.53 CO  co  I  KYmr  I  0.50  •:  0.40  I  A  KYR  I o.n • 0.09  o.o7  0.32  ;  |  0.24  : 0.05  1.38  I  0.18  ' 0.04 '  0.06  0.36  !  0.12  0.03  0.04  0.10  |  0.08  ;  0.02  |  0-02  0.24  ; 0.04  :  0.01  j  0.01  1.21  i.  0.003  !  0.00  I  12.68  j  j  j  0.02 0.005 0.00  ; 0.004 ;  0.001  ',  0.00  t r m -  ff  IT  i  •• r  -I.).,). |,|_  -H  J-U  TXT  •iT:" 1  7T7Tj[+  40.  • V a r i a n c e s ' oi j the j L s timator s! X. r s HH+rt when : Gamma : p j ) $ t t i b u t e d i P(x).; i s : f:*poneptia H - M - {-  H-HH-H  u  k  N 0.5 44-H  " • =;0. • To i l l u s t i a t e - T a b i e 1i 1 si r n t~l n ~n i :-rt-n- Ei : . ; : ! - : ! - | i  ± iurn.:::.;  :  T  TABLE V I I I V a r i a n c e s of the E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P(x) i s E x p o n e n t i a l , k = 0.05, g = 1. (Each e n t r y should be m u l t i p l i e d b y ^ / n ) I i  r = 1  r = 2  :  1  i  -i.o  ! HYHT 1 oo  j  -0.8  J141.30  ;  C  0.0  .  MYR  MYHT  KYmr  0.50  oo  0.50  0.25  oo  0.45  290.30  0.45  0.23  713.50  19.51  0.40  22.20  0.40  0.20  !  6.11  0.35  5.08  0.35  :  2.53  0.30  1.60  j 1  1 i  A  KYHT  •  \ -0.4 " -0.2  A  A  MYR  !  • -0.6  A  r = 3 j Hymr Ii A  t  A UYR  0.25 | 0.17 j  ; j  i  \ j  0.15  j  30.93  0.20 | 0.13  |  0.18-  5.33  0.18 | 0.12  i  0.30  0.15  1.47  0.15  0.10  |  0.23  j  -  :' 1.25  0.25  0.63  0.25  0.13  0.42  0.13  ;  0.66  0.20  0.27  0.20  0.10  0.16  0.10 ; 0.067 ;  0.4  .0.43  0.15  0.25  0.15  0.075  !  0.39  0.08 1 0.050 ';  0.6  0.47  0.73  0.10  0.050  j  1.50  0.05 i  0.033  3.54  0.05  0.025  0.03  0.017 ;  oO  0.00  0.00  0.2  ! 0.10 I  ; o.s  1  ; i.o  ! Var  1.10 ! 0.05 <?0  ;  0.00  (KYmr) i s n o t d e f i n e d  for r =  1.  11.15  j  cO  j  0.083. |  i  i  j  0.00 ' 0.00  TABLE I X V a r (KYHT) a n d v a r (KYmr) when X i s Gamma D i s t r i b u t e d , P ( x ) i s E x p o n e n t i a l , k = 0.5, g = 2. (.Each e n t r y s h o u l d be m u l t i p l i e d byk^/n) 1 1  • ' • — • • •-— •  1 <=  r  i. - 1  1  r = 2  r  t  KYHT  I  KYHT  KYmr  1 KYHT ;  A  A  KYmr  i  1  0.25  oo  165.67  0.25  15.13  -0.4  0.25  oo  301.10  0.25  720.6  0.25  24.20  0.25  32.06  j  7.00  0.25  5.79  0.25  5.81  j  -0.2  2.91  0.25  2.05  0.25  1.73  i 0.0  1.50  0.25.  0.88  0.25  0.67  0.74  0.25  0.47  0.25  0.42  0.46  0.25  0.46  | -1.0 t  j -o.s • ' ! -0.6  j  i 1 1  j  0 , 2  0.4  j 1.0  i  | I  0.25  1.06  j  oO  1  | i  0.25  |  0.25  |  0.71  1.00  |  0.25  |  - 2.26  '4.13  j  o o  |  0.25  j  0.25 0.25  ! |  | i  . 0.46  0.6  : o.s i  j 1  j  j j ; 1  i  ji 1  f  3  =  14.97 !  j  A  K Ymr  i  0.25  !  0.25  j  0.25 0.25  j  |  j  0.25 •  1  0.25  I  0.25'  !  0.25  j  0 25  |  0.25  |  o  i  0.25  ! !  • rl t t -tr  45. R e s u l t s f o r P ( x ) E x p o n e n t i a l and k = 1. The  r e s u l t s a r e g i v e n i n T a b l e s X - X I I and  i l l u s t r a t e d by F i g u r e s 10-12. The throughout.  r a t i o e s t i m a t o r has t h e s m a l l e s t v a r i a n c e When g = 0, r = 3 and 0.23<c o r c^-0.1, t h e  v a r i a n c e of t h e m e a n - o f - t h e - r a t i o s e s t i m a t o r i s s m a l l e r t h a n t h a t of t h e HT e s t i m a t o r .  The HT e s t i m a t o r b e a t s  t h e m e a n - o f - t h e - r a t i o s e s t i m a t o r f o r 0.1^c<0.28. When g = 1, r = 2 and c^O.12 or O 0 . 3 , t h e meano f - t h e - r a t i o s e s t i m a t o r i s more e f f i c i e n t t h a n t h e HT e s t i m a t o r , which beats the mean-of-the-ratios e s t i m a t o r f o r 0.12£c^0.3.  When r = 3 t h e m e a n - o f - t h e - r a t i o s  e s t i m a t o r has s m a l l e r v a r i a n c e t h a n t h e HT e s t i m a t o r . When g = 2 t h e v a r i a n c e of t h e m e a n - o f - t h e - r a t i o s e s t i m a t o r i s c o n s t a n t a t A / n a n d i s s m a l l e r t h a n t h a t of 2  the HT e s t i m a t o r .  I n t h i s p a r t i c u l a r case t h e minimum  p o i n t of t h e graph of t h e v a r i a n c e of HT e s t i m a t o r moves t o t h e l e f t and t h e v a l u e of t h e v a r i a n c e a t i t s minimum i n c r e a s e s as r i n c r e a s e s .  An i n c r e a s e i n t h e v a l u e s of g  has e f f e c t s s i m i l a r t o t h o s e mentioned i n t h e o t h e r above.  cases  As w i t h t h e p o l y n o m i n a l c a s e , an i n c r e a s e i n t h e  v a l u e of k f r o m 0.5 t o 1 has t h e e f f e c t of making t h e e s t i m a t o r s l e s s e f f i c i e n t f o r f i x e d v a l u e s of c, r and g.  TABLE X Variances P(x)  of the E s t i m a t o r s when X i s Gamma D i s t r i b u t e d ,  i s E x p o n e n t i a l , k = 1, g = 0.  (Each entry should be  m u l t i p l i e d by L( /n) 2  •  1 1  r = 1  1  i  it  C  A  I KYHT  rt  ;  RYR  r = 2  j  i  3.30  •  jf  :  30.80  1.28  0.28  5.13  0.49  i  5.25  0.98  0.22  |  1.77  0.36  1.41  0.72  0.16  2.56  ! 288.70 j 22.13  1.96  j  1.44  1  i  0.0  '  2.00  | 1.00  |  • 0.75  0.2  i  1.50  j !  0.64  !  0.4  !  1.44  0.36  0.6  |  1.77  0.8  '  3.53  1.0  j  oO  :  i  713.4  : 0.25  ;  0.31  0.50  0.11  0.45  0.16  i  0.23  0.32  0.07  j  0.57  0.09  0.52  0.18  0.04  0.16  !  1.41  0.04  ;  2.07  0.08  0.018  . 0.04  i  6.29  0.01  i  15.58  0.02  0.0044  0.00  0.00  :  0.00  oO  .0.00  oO  i  Var  A  H YR  0.64  j  -0.2  A  0.36  j  5.6i ;  : 1.00  !  A  1.62  |  1 t  oo  "  i  -0.6  i  A  !  0.81  -0.8  i  !  j {AYR  i  0.47  j  -0.4  A  MYHT 1  4.00 |i | 3.24  20.10  i  • KYHT —-1: KYmr ——— cQ ! 2.00  1 oo i j | 140.7  -1.0  r = 3  |  ( A Y mr)  i s not  defined for  k = l,  g = 0,  r c 2.  i  '  TABLE XI V a r i a n c e s of the E s t i m a t o r s when X i s Gamma D i s t r i b u t e d , P(x) i s E x p o n e n t i a l , k = 1, g = 1. (Each e n t r y should be m u l t i p l i e d by jj^/n)  r - 1 A  c  KYHT  AYR  A.  A  KYHT  K.Ymr  2.00  -1.0 -0.8  r - 2  151.8  •  i  KYR  r =3  A  A  A  KYHT  KYmr  KYR  2.00  | 1.00  o O  1.00  0.67  1.80  292.20  1.80  ! 0.90  714.80  0.90  0.60  -0.6  22.44  1.60  59.55  1.60  ; 0.80  31.30  0.80  0.53  -0.4  7.60  1.40  5.61  1.40  j  0.70  5.59  0.70  0.47  -0.2  3.56  1.20  2.25  1.20  : o.60  1.70  0.60  0.40  0.0  2.00  1.00  2.00  1.00  ! 0.50  0.67  0.50  0.33  0.2  1.31  0.80  0.67  0.80  0.50  0.40  0.27  1.07  0.60 |  0.78  0.60  0.98  0.30  0.20  1.20  0.40 |  1.64  0.40  3.03  0.20  0.13  0.10  0.07  0.00  0.00  i  j 1 1  0.4 0.6 0.8  2.26 oO  i  0.00  • 1.0  A  Var  6.43  0.20  j  C O  0.20 0.00  j 0.40 j 0.30 1 0.20 i  1 °- ,  10 20.79  > o oo ; 0  1  (MYmr) i s not d e f i n e d f or. r = 1.  oO  iTf  4-4-4 I 1 1 I I 1HJ 4I III I U J-U-iJ-Hri Vaxianc es I of I the i Stima tor Gariiriia ; D i c t x i b i j i t e d P ( x ) m t m :  19.  :: R± it  - h  r~n~|  • FFr  i  A  A  var  TABLE XII  (KYHT) and v a r (KYmr) when X i s Gamma D i s t r i b u t e d ,  P(x)  i s E x p o n e n t i a l , k = 1, g = 2.  (Each entry  should  be m u l t i p l i e d by H^/n)  1  = i  i  r = 2 A  A  c  ;  &YHT  - 1.0  i  - 0.8  "' 276.78  - 0.6  r = .3  UYHT  •UYrnr  cO  OO  1  $Ymr  KYHT  $ Ymr  oO  1  1  742.0  1  346.30  1  • 37.46  .1  31.04  1  37.87  1  - 0.4  12.16  1  8.95  1  7.50  1  - 0.2  5.51  1  3.58  1  2.86  1  0.0  3.00  2.00  1  1.67  1  0.2  1.90  1.49  1  1.55  1  0.4  1.43  I  1.60  1  2.40  1  0.6  1.45  |  2.72  1  6.08  1  0.8  2.51  9.00  1  oo  1  1.0  oO i  i  1  i  !  i  !  1  1 i  ! 1  i  1  1  ! !  i  |  36.41  1  |  1  i  1  52.  Summary of R e s u l t s and Some The  Conclusions  f o l l o w i n g o b s e r v a t i o n s can be made from the  results: (1)  Under the g i v e n assumptions about the d i s t r i b u t i o n of the a u x i l i a r y v a r i a b l e , X,and the sampling and  designs,  f o r g = 0 or 1, the r a t i o e s t i m a t o r i s b e t t e r than  both the m e a n - o f - t h e - r a t i o s  and the HT e s t i m a t o r s .  Under a very wide range of the v a l u e s of m, the meano f - t h e - r a t i o s e s t i m a t o r performs b e t t e r than the HT estimator. (2)  The HT e s t i m a t o r i s v e r y  s e n s i t i v e t o changes i n  v a l u e s of the d e s i g n f u n c t i o n parameters m and c. Within a c e r t a i n  (small) range of the v a l u e s of these  parameters the HT e s t i m a t o r has, sometimes, smaller v a r i a n c e than t h a t of the m e a n - o f - t h e - r a t i o s (3)  estimator.  The v a r i a n c e s of these e s t i m a t o r s are u s u a l l y smaller when k = 0„5 than when k = 1.  (4)  For f i x e d v a l u e s of k and  g, the v a r i a n c e s of the  r a t i o and m e a n - o f - t h e - r a t i o s i n c r e a s e w i t h a decrease  estimators usually  i n the value  of r .  With the  HT e s t i m a t o r , i t depends on the range of v a l u e s of m or c c o n s i d e r e d and the d e s i g n f u n c t i o n used. the polynominal  With  d e s i g n f u n c t i o n , the e f f i c i e n c y of  the HT e s t i m a t o r u s u a l l y i n c r e a s e s w i t h r .  With the  e x p o n e n t i a l d e s i g n f u n c t i o n and f o r , say, c<0.3, the  53.  efficiency  of the HT e s t i m a t o r improves with an  i n c r e a s e i n the value of r ; when c>0.4 the HT e s t i m a t o r u s u a l l y becomes l e s s e f f i c i e n t as r takes higher v a l u e s . For the same v a l u e s of k, and f o r the same ranges of the v a l u e s of the d e s i g n f u n c t i o n m or c, the e f f e c t of i n c r e a s i n g g  parameters, i s usually to  make the r a t i o and the m e a n - o f - t h e - r a t i o s e s t i m a t o r s less e f f i c i e n t . Other t h i n g s f i x e d , the v a r i a n c e s of the r a t i o and the m e a n - o f - t h e - r a t i o s e s t i m a t o r s decrease w i t h an i n c r e a s e i n the v a l u e s of m or c. The r a t i o and m e a n - o f - t h e - r a t i o s e s t i m a t o r s have s m a l l e r v a r i a n c e s when s e l e c t i o n (m?0  i s 'purposive'  or c70) than when simple random  sampling  (m=c=0) i s employed. The HT e s t i m a t o r i s most e f f i c i e n t when the sampling scheme i s approximately  pps, i . e . when m i s  approximately = 1 ( i n the polynominal c a s e ) . In most c a s e s , the v a r i a n c e s of these e s t i m a t o r s can be e q u a l i z e d by simply choosing combinations  of the parameters.  different  For example, the  v a r i a n c e of the r a t i o e s t i m a t o r f o r k = 1, g = 1 and r = 1 e q u a l s the v a r i a n c e ofthe  mean-of-the-ratios  e s t i m a t o r w i t h k = 1, g = 1 and r = 2 (and P(x) i s polynominal).  54. (10)  F o r a g r e a t e r range of the v a l u e s  o f c o r m, t h e  e s t i m a t o r s have s m a l l e r v a r i a n c e s when P ( x ) polynominal (11)  t h a n when i t i s e x p o n e n t i a l .  W i t h an e x p o n e n t i a l d e s i g n  function,  of t h e e s t i m a t o r s a r e f i n i t e range of t h e v a l u e s ratio  of c.  and m e a n - o f - t h e - r a t i o s  when c = 0.  i s  the variances  f o r a very  The v a r i a n c e s  small of t h e  estimators are zero  55.  CHAPTER 4 DISCUSSION OF  RESULTS  Some E m p i r i c a l Comparisons P.S.R.S. Rao  (1969) compared f o u r r a t i o - t y p e  e s t i m a t o r s under the r e g r e s s i o n model: y=(X+p>X+e  where  E(e.| ) = 0 X i  E(  e.  e i  Var  | X  ±  (ej_ j X ) L  Xj) = 0  =  £x?  and X i s gamma d i s t r i b u t e d w i t h parameter h. out t h a t when a = 0 and  He  found  g > 1, the MSE's i n c r e a s e as h  increases.  He  s i z e 2-10.  These r e s u l t s are not q u i t e compatible  my  c o n s i d e r e d v a l u e s of h>2  r e s u l t s f o r the  f o r g>2,  But  (Rao's f i r s t  from (2.5)  and  estimator  For g  i s the  with  ratio  f o r g > 1, I do not have much to compare w i t h .  as r i n c r e a s e s .  f i x e d n and  Rao  a l s o found  h (or r ) the MSE's i n c r e a s e as  T h i s agrees with my (2.7).  estimator  2, the v a r i a n c e decreases  C e r t a i n l y when g = 1, the v a r i a n c e of the r a t i o decreases  ratio  (2.9), i t i s c l e a r t h a t ,  Since I d i d not work out the v a r i a n c e of the  estimator  with  mean-of-  the v a r i a n c e of the m e a n - o f - t h e - r a t i o s  increases with r . r.  samples of  same MSE's of the r a t i o and  the-ratios estimators estimator).  and  r e s u l t s and  can be  estimator  out t h a t f o r g  increases.  i n f e r r e d from (2.5)  L a s t l y , h i s r e s u l t s show t h a t f o r 0f= 0,  n 72  and  -  56. g = 1 or 2, the r a t i o e s t i m a t o r has s m a l l e r MSE than the other t h r e e .  I t i s i m p l i e d from h i s paper t h a t a c h o i c e  of a and g.combinations does a f f e c t the MSE's of the estimators. My r e s u l t s and  show t h a t the v a r i a n c e s of the r a t i o  mean-of-the-ratios  e s t i m a t o r s become s m a l l e r as the  d e s i g n f u n c t i o n parameter m or c i n c r e a s e s . (2.7),  (2.10) - (2.11) show s i m i l a r r e s u l t s .  (2.5) But the  l a r g e r the v a l u e s of m or c the h i g h e r w i l l be the i n c l u s i o n p r o b a b i l i t i e s f o r u n i t s w i t h l a r g e v a l u e s of X compared w i t h those w i t h  s m a l l v a l u e s of X.  T h i s leads  t o the n o t i o n t h a t i f u n i t s w i t h l a r g e v a l u e s of purposely  i n c l u d e d i n the sample, the e s t i m a t i o n  w i l l be more e f f i c i e n t . (1970, 1971) have found conditions.  X  procedure  Many r e s e a r c h e r s , n o t a b l y , R o y a l l s i m i l a r r e s u l t s under  similar  R o y a l l ' s r e s u l t s a l s o show t h a t the r a t i o  e s t i m a t o r i s b e t t e r when combined w i t h d e s i g n s other simple random sampling.  it  When used w i t h  the non-existence  unbiased  Since  design, Godambe  .of a u n i f o r m l y minimum v a r i a n c e  e s t i m a t o r among a c l a s s of a l l unbiased  e s t i m a t o r s f o r any sampling note  unbiased  simple random sampling  remains o p t i m a l only i f M i s i n f i n i t e .  proved  than  For g = 1, k = 1, he shows t h a t  the r a t i o e s t i m a t o r i s the b e s t l i n e a r i estimator.  are  linear  d e s i g n , i t may be proper t o  t h a t w h i l e d i s c u s s i n g these  o p t i m a l i t y p r o p e r t i e s of  the r a t i o e s t i m a t o r we a r e , most of the time,  restricting  57. o u r s e l v e s t o a c e r t a i n c l a s s of l i n e a r (Godambe's 1955  class  (iii)  ).  The  estimators  same remarks apply  to o p t i m a l i t y c o n d i t i o n s g i v e n by Cochran HT  only  (1963).  The  e s t i m a t o r belongs t o Godambe's (1955) s u b - c l a s s ( i )  of e s t i m a t o r s .  I t i s the best and  l i n e a r estimator  i n the  set up,  the problem of choosing  the Midzumo-Sen sampling  combined w i t h  Rao  and  the HT  other two  the HT  strategy estimators.  scheme.  My  mean-of-the-ratios  simple  results  suggest  estimators  random sampling,  with that  are  t h e r e always e x i s t s mean-of-the-ratios  i s l e s s than the v a r i a n c e s of the r a t i o and  estimators.  S i m i l a r l y , the v a r i a n c e  of the  e s t i m a t o r w i t h v a r y i n g p r o b a b i l i t y of i n c l u s i o n can be made s m a l l compared w i t h those o f - t h e - r a t i o s estimators with If  estimator  e s t i m a t o r s used  an m or c such t h a t the v a r i a n c e of the estimator  (1967)  t h a t the m e a n - o f - t h e - r a t i o s  w i t h ITPs i s b e t t e r than the  i f the r a t i o and  unbiased  a suitable  f o r the r a t i o , m e a n - o f - t h e - r a t i o s H i s r e s u l t s suggest  only  sub-class.  Under s u p e r - p o p u l a t i o n considered  the  i n formula  (2.2)  simple  f o r the  ratio  procedure  of the HT and random  of  mean-  sampling.  mean-of-the-ratios  e s t i m a t o r , we l e t *i •r the HT  estimator  =  (2.1)  P(Xi) i s obtained.  Hanurav  (1967)  was  58  o  i n t e r e s t e d i n f i n d i n g sampling d e s i g n s under which t h i s X• p(X-) = — and the v a r i a n c e of the r n e a n - o f - t h e - r a t i o s r 1  e s t i m a t o r i s unbiased total).  (he was e s t i m a t i n g the p o p u l a t i o n  For n = 2, he gives,two  procedures t h a t  s e q u e n t i a l sampling  s o l v e the problem.  With my study, the  c o n d i t i o n s a r e easy t o f i n d : when  P(X) P(Xi)  when  i s p o l y n o m i n a l , we want = X^ r  • (m+ r j  P(X)  i s e x p o n e n t i a l we have  P(Xi)  ( 1  - c)  r  cxi X i r =  e  I t i s easy t o f i n d an r (c or m) t h a t w i l l  g i v e the .  r e q u i r e d P(Xj_) f o r f i x e d m or c (or r ) p r o v i d e d c ~ m = 0. With polynominal P(x), t h i s  study i s r e a l l y a  s p e c i a l case of t h a t c o n s i d e r e d by C a s s e l and S a r n d a l (1973). N e a r l y a l l of my f i n d i n g s a r e i n agreement w i t h t h e i r findings.  They f i n d  out, f o r example, t h a t :  ( i ) When g = 2, the m e a n - o f - t h e - r a t i o s e s t i m a t o r has c o n s t a n t v a r i a n c e r e g a r d l e s s of b o t h the d e s i g n and the d i s t r i b u t i o n f u n c t i o n , f ( x ) . ( i i ) The r a t i o e s t i m a t o r combined w i t h simple random sampling if (iii)  can be q u i t e i n e f f i c i e n t  used w i t h other sampling  compared w i t h i t  designs,  Under c e r t a i n c o n d i t i o n s , one should use purposive s e l e c t i o n of the u n i t s with l a r g e s t  X-values.  59.  ( i v ) When g ^'1 the r a t i o e s t i m a t o r i s at l e a s t as efficient  as the m e a n - o f - t h e - r a t i o s  any d e s i g n P ( x ) . ing,  estimator f o r  In the case t h a t I am c o n s i d e r -  f o r n>l and g ^.1, the e x p r e s s i o n s f o r the  v a r i a n c e s of the r a t i o and the m e a n - o f - t h e - r a t i o s e s t i m a t o r s show t h a t the r a t i o e s t i m a t o r has s m a l l e r v a r i a n c e than the m e a n - o f - t h e - r a t i o s e s t i m a t o r , and the d i f f e r e n c e between the two v a r i a n c e s i n c r e a s e s with n.  For example, when  g = 0 and P(x) i s polynominal,  we have A  nk  (M-Ymr) =  Var  n (m+r-l)(m+r-2) 2  2  V  a  r  (  (v) V a r i a n c e s  ^  Y  R  )  =  2  Hy  2  it  2  (nm nr-l)(nm+nr-2 +  of the r a t i o and  mean-of-the-ratios  e s t i m a t o r s i n c r e a s e w i t h m or c . ( v i ) I f g = 2, r e g a r d l e s s of the d e s i g n f u n c t i o n used,  '  the m e a n - o f - t h e - r a t i o s  e s t i m a t o r has v a r i a n c e t h a t  i s s m a l l e r than,  a l t o t h a t of the r a t i o  estimator  (I d i d not work out the v a r i a n c e of the  r a t i o e s t i m a t o r when g = 2 ) . (vii) If  g > 2 the v a r i a n c e of the m e a n - o f - t h e - r a t i o s  estimator, becomes small i f the d e s i g n a s s i g n s the bulk of the s e l e c t i o n p r o b a b i l i t i e s with  smallest  X-values.  to the u n i t s  60. ( v i i i ) The HT e s t i m a t o r shifts  i s generally highly sensitive to  i n the d e s i g n .  I t s variance  i s a minimum'  when s e l e c t i o n p r o b a b i l i t i e s are somewhat i n the. vicinity  of pps procedure.  The v a r i a n c e can  become v e r y l a r g e due t o minor d e v i a t i o n s from the p o i n t of minimum. Comments and Some I m p l i c a t i o n s From the r e s u l t s of t h i s  study,  i t would seem  t h a t i n s t e a d of t r y i n g t o look f o r e s t i m a t o r s t h a t are generally  o p t i m a l l i k e u n i f o r m l y minimum v a r i a n c e  more e f f o r t  estimators,  should be used i n d e f i n i n g c l e a r l y and simply  the c o n d i t i o n s under which t h e popular  e s t i m a t o r s are  efficient. Under the u s u a l r e g r e s s i o n models, the c h o i c e of g,  i t seems, determines the extent t o which the best  c h o i c e of sampling process. value  s t r a t e g y can improve the e s t i m a t i o n  In most, cases  of g  s t u d i e d , i t has been found  l i e s between 1 and 2.  T h i s may be  unfortunate  as the t h r e e common e s t i m a t o r s I have c o n s i d e r e d I  more e f f i c i e n t when g = 0.  t h a t the  can be  On the other hand, g = 0  i m p l i e s t h a t the v a r i a n c e o f the e r r o r term, Zj_, i s constant which i s a v e r y u n l i k e l y For 1 £ g £ 2 ,  situation i n practice.  the v a r i a n c e s of the t h r e e e s t i m a t o r s can be  made t o a t t a i n t h e i r minimum v a l u e s by a proper rn, r and g.  Perhaps the good t h i n g with the  c h o i c e of  continuous  61. v a r i a b l e model i s t h a t the e x p r e s s i o n s f o r t h e ' v a r i a n c e s of  the e s t i m a t o r s are very, v e r y  simple.  I f one was  i n t e r e s t e d , i n g e t t i n g the exact minimum v a l u e s of the v a r i a n c e s of these e s t i m a t o r s , i t should not be t o o difficult  f o r him t o do so.  He w i l l  likely  have t o use  the computer and some mathematical programming The  r e s u l t s a l s o c a l l f o r more a t t e n t i o n t o the  c h o i c e of e s t i m a t o r s and sampling 1  survey  techniques.  sampling.  d e s i g n s when doing  In p a r t i c u l a r , the r e s u l t s show a g a i n ,  1 t h a t i n most cases, optimal And  sampling  simple random sampling  design.  i s not an  There are other b e t t e r ones.  when e s t i m a t i n g the p o p u l a t i o n mean, the sample average  need not g i v e optimal r e s u l t s .  There may be other b e t t e r  estimators. In p r a c t i c e , the sampler, under t h i s model, w i l l p a r t l y be a b l e t o c o n t r o l the d e s i g n f u n c t i o n parameters. In such  situations,  make a proper will  s t u d i e s l i k e t h i s may h e l p the sampler  c h o i c e of the d e s i g n f u n c t i o n parameter t h a t  give b e s t r e s u l t s .  In t h i s  study, the r a t i o and  rtiean-of-the-ratios e s t i m a t o r s , as i n s i m i l a r other s t u d i e s , promise good r e s u l t s when s e l e c t i o n i s p u r p o s i v e certain conditions.  Most sample survey  under  experts object t o  t h i s method of s e l e c t i o n because, as Hansen, Hurwitz and Madow (1953, p.9) put i t : (a) Methods of s e l e c t i n g samples based on the theory of p r o b a b i l i t y are the only g e n e r a l methods known to us which can p r o v i d e a measure of p r e c i s i o n .  62. Only by using p r o b a b i l i t y methods can o b j e c t i v e numerical statements be made c o n c e r n i n g the p r e c i s i o n of the r e s u l t s of the survey; (b)  I t i s necessary t o be sure t h a t the c o n d i t i o n s imposed by the use of p r o b a b i l i t y methods are satisfied. I t i s not enough t o hope or expect t h a t they a r e . Steps must be taken t o meet these c o n d i t i o n s by s e l e c t i n g methods t h a t are t e s t e d and are demonstrated to conform to the p r o b a b i l i t y model.  They c o n t i n u e  saying:  We a s s e r t t h a t , w i t h r a r e e x c e p t i o n s , the p r e c i s i o n of e s t i m a t e s not based on known proba b i l i t i e s of s e l e c t i n g the samples cannot be p r e d i c t e d b e f o r e the survey i s made, nor can the p r o b a b i l i t i e s or p r e c i s i o n be e s t i m a t e d a f t e r the sample i s o b t a i n e d . I f we know nothing of the p r e c i s i o n , then we do not know whether t o have much f a i t h i n our e s t i m a t e s , even though h i g h l y a c c u r a t e measurements are made on the u n i t s i n the sample. Random sampling  i s usually  supported  namely, i t p r o t e c t s a g a i n s t f a i l u r e assumptions,  i t averages  unknown random v a r i a b l e s ,  for similar of c e r t a i n  arguments,  probabilistic  out e f f e c t s of unobserved or i t guards a g a i n s t  b i a s on the p a r t of the experimenter, produce a sample i n which the X's  i t will  unconscious usually  are spread throughout  range of X v a l u e s i n the p o p u l a t i o n and t h i s enables sampler to check t h e . a c c u r a c y r e l a t i o n of the y's t o the X's sampler t o e s t i m a t e , from the  the  of assumptions c o n c e r n i n g and,  a g a i n , i t enables  the  the  the  sample, the p r e c i s i o n of h i s  estimate. But  p r o b a b i l i t y methods can do nothing more than  give us e x p e c t a t i o n s about,  say, the p o s s i b l e p r e c i s i o n of  6 3 .  the r e s u l t s of the be  survey.  The  p r e c i s i o n would u s u a l l y  s t a t e d i n terms of the p r o b a b i l i t y t h a t the  estimate  d e v i a t e s from the r e a l v a l u e , and as long as i t i s g i v e n i n these p r o b a b i l i t y terms, i t does nothing e x p e c t a t i o n s , however h i g h and r e f i n e d the may  be.  On  the  more than  probabilities  other hand, i f many s t u d i e s p o i n t t o the  f a c t t h a t n o n - p r o b a b i l i t y methods, l i k e p u r p o s i v e l e a d , under c e r t a i n c o n d i t i o n s , to e f f i c i e n t the  same p r o b a b i l i t y theory may  sampling  c o n d i t i o n s , we  selection,  estimations,  a l l o w t h a t under  can expect, w i t h h i g h  t o o b t a i n s i m i l a r good r e s u l t s . i f the  give  similar  probability,  As R o y a l l (1970)  sampler b e l i e v e s i t t o be important  argues,  t h a t he  o b t a i n ..  a sample i n which the X v a l u e s have a c e r t a i n c o n f i g u r a t i o n , then he it  should choose a sample d e l i b e r a t e l y and  not  t o the c h o i c e of a c e r t a i n chance mechanism. In t h i s study, the r a t i o e s t i m a t o r has  once a g a i n  shown i t s u p e r i o r i t y over the m e a n - o f - t h e - r a t i o s estimators.  The  HT  design.  The  of the  r n e a n - o f - t h e - r a t i o s e s t i m a t o r may  o f f from the r a t i o The  the  HT  very  estimator.  r e s u l t s of the'study  to note t h a t s t r a t i f i e d  sampling  not be  are q u i t e s i m i l a r t o  s t u d i e s under d i f f e r e n t r e g r e s s i o n type models.  sampling  and  e s t i m a t o r has r e v e a l e d i t s s e n s i t i v i t y  t o the c h o i c e of parameters of the model and  far  leave  sampling  combined w i t h  i n each stratum can be achieved,  similar  I would simple  like  random  under t h i s model,  64. by a s s i g n i n g the same P(x) f o r a l l members of one stratum and d i f f e r e n t  P(x) f o r members of d i f f e r e n t  strata.  Some L i m i t a t i o n s In p u t t i n g the ideas- c o n t a i n e d i n t h i s study practice,  the order of events i s  distribution  ( l ) e s t i m a t e F ( x ) , the  f u n c t i o n of x (2) approximate  (3)"investigate  some sampling  of d i f f e r e n t  g and 9 and  d e s i g n s and e s t i m a t o r s and  choose the ones t h a t give b e s t r e s u l t s . h e l p the sampler  into  t o approximately  T h i s study would  examine the b e h a v i o u r s  d e s i g n s and e s t i m a t o r s he may be pondering  t o use. The  assumption  c o n c e r n i n g an i n f i n i t e p o p u l a t i o n  t h a t has a p p r o x i m a t e l y a c o n t i n u o u s frequency while i t helps s i m p l i f y  the i n v e s t i g a t i o n  d e s i g n s , a l s o makes the s i t u a t i o n of the r e a l of  situation.  distribution,  of d i f f e r e n t  c o n s i d e r e d an i d e a l i z a t i o n  To estimate the f r e q u e n c y  function  x, one c o u l d s t a r t by o b s e r v i n g the h i s t o g r a m of the x  v a l u e s and choose or f i t an approximate p o s s i b l y by some mathematical closely  resembles  continuous  function,  curve f i t t i n g t e c h n i q u e s , t h a t  the histogram;  and the c o n t i n u o u s  function  thus obtained has t o be s t a n d a r d i z e d t o become a cumulative distribution  function.  use some l i k e l i h o o d Brewer  (1963).  To approximate  methods l i k e  6 and g, we c o u l d  the ones suggested by  65. I t i s q u i t e p o s s i b l e t h a t some problems of e v a l u a t i n g the v a r i a n c e s of the e s t i m a t o r s we  may  want  t o i n v e s t i g a t e g i v e n the approximate d i s t r i b u t i o n f u n c t i o n of• x w i l l be encountered.  Should  i t , f o r example, t u r n  out t h a t the X v a l u e s are approximately and the sampler wants t o i n v e s t i g a t e the  normally  distributed  sampling  designs  and e s t i m a t o r s I have s t u d i e d , i t would not be easy evaluate the v a r i a n c e s . f u n c t i o n s and and  sampling  But w i t h other  easy.  distribution  d e s i g n s t h i n g s should be  studying the p r o p e r t i e s of such C a s s e l and S a r n d a l  sampling  easy  going  strategies i s  (1973) show t h a t r e s u l t s  under t h i s model are v a l i d  to  obtained  f o r v a l u e s of N as low as N =- 10.  Some Recommendations I t h i n k sample survey t h e o r i s t s should spend more time  s i m p l i f y i n g and  They should  u n i f y i n g the r e s u l t s of t h e i r r e s e a r c h .  spend more time  number of d i f f e r e n t  sampling  they are c o n s i d e r i n g .  i n r e f i n i n g and designs and  They should,  simple u n i f i e d theory of sampling into practice. considered may up  reducing  the  estimators that  somehow, formulate  t h a t can e a s i l y be  a  put  I n ' t h i s c o n n e c t i o n , the e s t i m a t o r s I have prove u s e f u l .  S u r e l y , some problems may  crop  initially. I a l s o t h i n k that some ideas from G e n e r a l  Theory can be u s e f u l i n f o r m u l a t i n g a simple theory; there i s no need of d i v o r c i n g one t h e o r i e s from the  other.  Statistical  sample  of the  survey  sampling  66.  REFERENCES Basu, D. (1971). An essay on the l o g i c a l f o u n d a t i o n s of survey sampling, P a r t 1. i n Foundations of S t a t i s t i c a l I n f e r e n c e , ed. by V. P. Godambe and D. A. S p r o t t , Toronto: H o l t , R i n e h a r t and Winston of Canada L t d . , 203-242. Brewer, K.R.W. (1963). R a t i o e s t i m a t i o n and f i n i t e populations: some r e s u l t s d e d u c i b l e from the assumption of an u n d e r l y i n g s t o c h a s t i c p r o c e s s . A u s t . J . S t a t i s t . 5. 93-105. C a s s e l , C M . and S a r n d a l , C.E. (1972a). A model f o r studying r o b u s t n e s s of e s t i m a t o r s and i n f o r m a t i v e ness of l a b e l s i n sampling w i t h v a r y i n g p r o b a b i l i t i e s , J . R. S t a t i s t . Soc. B., 35. 279-289. , (1972b). An a n a l y t i c framework o f f e r i n g some g u i d e l i n e s t o the c h o i c e of sampling d e s i g n and the c h o i c e of e s t i m a t o r f o r f i n i t e p o p u l a t i o n s . Working paper No. 142, F a c u l t y of Commerce, UBC. , (1973). E v a l u a t i o n of some sampling s t r a t e g i e s f o r f i n i t e populations using a continuous v a r i a b l e framework. To appear i n the A p r i l i s s u e of Communications i n S t a t i s t i c s . Cochran, W.G. (1946). R e l a t i v e accuracy of s y s t e m a t i c and s t r a t i f i e d random samples f o r a c e r t a i n c l a s s of p o p u l a t i o n s . Ann. Math. S t a t i s t . , 17, 164-177. , (1963).  Sampling  Techniques.  New York:  Wiley.  Durbin, J . , (1953). Some r e s u l t s i n sampling t h e o r y when the u n i t s are s e l e c t e d with unequal p r o b a b i l i t i e s . J . R. S t a t i s . Soc. B., 15. 262-269. Foreman, E.K., and Brewer, K.R.W. (1971). The e f f i c i e n t Use of Supplementary I n f o r m a t i o n i n Standard Sampling P r o c e d u r e s . J.R. S t a t i s t . Soc. B.. 33. 391-400. Godambe, V.P. (1955). A u n i f i e d theory of sampling from f i n i t e p o p u l a t i o n s . J.R. S t a t i s t . S o c . B . , 17, 269-278. , (1965). A review of the c o n t r i b u t i o n s towards a u n i f i e d t h e o r y of sampling from f i n i t e p o p u l a t i o n s . Rev. I n t . S t a t i s t . I n s t . . 32.. 242-253.  67. Godambe, V.P. (1969). Some a s p e c t s of the t h e o r e t i c a l developments i n survey sampling. New Developments i n Survey Sampling, ed. by N. L. Johnson and H. Smith, J r . , 27-58. New York: Wiley. ., (1970). Foundations of Survey-sampling. American S t a t i s t i c i a n , 24. 33-38.  The  and J o s h i , V.M. (1965). A d m i s s i b i l i t y and Bayes e s t i m a t i o n i n sampling f i n i t e p o p u l a t i o n s , 1 . Ann. Math. S t a t i s t . . . 36. 1707-1722. Goodman, R. and K i s h , L. (1950). C o n t r o l l e d s e l e c t i o n - A technique i n p r o b a b i l i t y sampling. J . Amer. S t a t i s t , A s s o c . . 45. 350-372. Hansen, M.H., Hurwitz, W.N. and Madow, W.G. (1953). Sampling Survey Methods and i h e o r y , V o l . 11. Wiley & Sons, New York.  John  Hanurav, T.V. (1967). Optimum u t i l i z a t i o n of A u x i l i a r y Information: f/"PS sampling of two u n i t s from a stratum. J . R. S t a t i s t . Soc. B. 29, 374-391. H a r t l e y , H.O. and Rao, J.N.K. (1962). Sampling w i t h unequal p r o b a b i l i t i e s and without replacement. Ann. Math. S t a t i s t . 2. 77-86. H o r v i t z , D.G. and Thompson, D.J. (1952). A g e n e r a l i z a t i o n of sampling without replacement from a finite u n i v e r s e . J . Amer. S t a t i s t . A s s . 47. 663-635. J o s h i , V.M. (1969). A d m i s s i b i l i t y of E s t i m a t e s of the mean of a f i n i t e p o p u l a t i o n . New Developments i n Survey Sampling, ed. by N. L. Johnson and H. Smith, J r . , 188-207, New York: Wiley. Neyman, J . (1934). On the two d i f f e r e n t a s p e c t s of the r e p r e s e n t a t i v e method. The method of s t r a t i f i e d sampling and the method of p u r p o s i v e s e l e c t i o n . J . R. S t a t i s t . Soc. 97.. 558-606. Rao,  J.N.K. (1969). R a t i o and R e g r e s s i o n E s t i m a t o r s , New Developments i n Survey Sampling. Ed. by N.L. Johnson and H. Smith, J r . , 213-234, New York, Wiley.  Rao,  P.S.R.S. (1969). Comparison of four r a t i o - t y p e e s t i m a t e s under a model. J . Amer. S t a t i s t . A s s o c . 64. 574-580.  oo. Rao, T . J . (1967). On the choice of a s t r a t e g y f o r the R a t i o Method of e s t i m a t i o n . J . R. S t a t i s t . Soc. B. 29, 392-397. R o y a l l , R.M. (1970). On f i n i t e p o p u l a t i o n sampling theory under c e r t a i n l i n e a r r e g r e s s i o n models. B i o m e t r i k a , 57. 377-387. , (1971). L i n e a r r e g r e s s i o n models i n f i n i t e p o p u l a t i o n sampling t h e o r y . In Foundations of S t a t i s t i c a l I n f e r e n c e , ed. by V.P. Godambe and D. A. S p r o t t , Toronto: H o l t , R i n e h a r t and Winston of Canada L t d . , 259-279. S a r n d a l , C.E. (1972). Sample Survey theory V s . G e n e r a l S t a t i s t i c a l theory: e s t i m a t i o n of the p o p u l a t i o n mean. Rev. I n t . S t a t i s t . I n s t . , 40. 1-12. Sukhatme, P.V. and Sukhatme, B.V. (1970). Sampling Theory of Surveys w i t h A p p l i c a t i o n s , A s i a P u b l i s h i n g House, Bombay 1.  

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