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Parameter estimation in some multivariate compound distributions 1965

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PARAMETER ESTIMATION IN SOME MULTIVARIATE COMPOUND DISTRIBUTIONS GEORGE E. J . SMITH B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics We accept t h i s t h e s i s as conforming t o the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1965 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that per- mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives* It is understood that copying or publi- cation of this thesis for financial gain shall not be allowed without my written permission. Department of ^J^til^Ag^U^^> The University of British Columbia Vancouver 8, Canada Date i i ABSTRACT During the past three decades or so there has been much work done concerning contagious p r o b a b i l i t y d i s t r i b u t i o n s i n an attempt t o e x p l a i n the behavior of c e r t a i n types of b i o l o g i c a l p o p u l a t i o n s . The d i s t r i b u t i o n s most widely discussed have been the P o i s s o n - b i n o m i a l , the Poisson P a s c a l or Poisson-negative binomial,' and the Poisson-Poisson or Neyman Type A. Many ge n e r a l - i z a t i o n s of the above d i s t r i b u t i o n s have a l s o been discussed. The purpose of t h i s work i s t o d i s c u s s the m u l t i v a r i a t e analogues of the above three d i s t r i b u t i o n s , i . e . the Po i s s o n - m u l t i n o m i a l , Poisson-negative m u l t i n o m i a l , and P o l s s o n - m u l t i - v a r i a t e P o i s s o n , r e s p e c t i v e l y . I n chapter one the f i r s t of these d i s t r i b u t i o n s i s discussed. I n i t i a l l y a b i o l o g i c a l model i s suggested which leads us t o a p r o b a b i l i t y generating function. Prom t h i s a r e c u r s i o n formula f o r the p r o b a b i l i t i e s i s found. Parameter e s t i m a t i o n by the methods of moments and maximum l i k e l i h o o d i s discussed i n some d e t a i l and an approximation f o r the asymptotic e f f i c i e n c y of the former method i s found. The l a t t e r method i s a s y m p t o t i c a l l y e f f i c i e n t . F i n a l l y sample zero and u n i t sample frequency e s t i - mators are b r i e f l y d i s c u s s e d . I n chapter two, e x a c t l y the same procedure i s fol l o w e d f o r the Poisson-negative m u l t i n o m i a l d i s t r i b u t i o n . Many clo s e s i m i l a r i t i e s are obvious between the two d i s t r i b u t i o n s . i i i The l a s t chapter i s devoted t o a p a r t i c u l a r common l i m i t i n g case of the f i r s t two d i s t r i b u t i o n s . T h i s i s the P o l s s o n - m u l t i - v a r i a t e Poisson. I n t h i s case the d e s i r e d r e s u l t s are obtained by c a r e f u l l y c o n s i d e r i n g appropriate l i m i t s i n e i t h e r of the previous two cases. i v TABLE OF CONTENTS page INTRODUCTION 1 CHAPTER I The Poisson M u l t i n o m i a l D i s t r i b u t i o n 1-1. A B i o l o g i c a l Model 2 1-2. P r o b a b i l i t y Generating Function and Recursion Formula f o r P r o b a b i l i t i e s . . 4 1-3. E s t i m a t i o n of Parameters by the Method of Moments 6 1-4. Maximum L i k e l i h o o d Estimators . . 10 1-5- Covariance M a t r i x of Maximum L i k e l i h o o d E s t i mators A. Method of C a l c u l a t i o n . . . . 16 B. C a l c u l a t i o n of the elements of J . . 2 2 1-6. E f f i c i e n c y of Method of Moments A. Method of C a l c u l a t i o n . . . . 26 B. C a l c u l a t i o n of 'det M' i n Terms of the Parameters 28 C. Determination of the Jacobian 'det (J' 32 1-7. Sample Zero Frequency and U n i t Sample Frequency Estimators A. Sample Zero Frequency and F i r s t Moments 3^ B. U n i t Sample Frequency E s t i m a t i o n 36 page CHAPTER I I The Poisson Negative-Multinomial D i s t r i b u t i o n 2-1. A B i o l o g i c a l Model .. . . . . . 38 2-2. P r o b a b i l i t y Generating Function and Recursion Formula f o r P r o b a b i l i t i e s . . 39 2-3. E s t i m a t i o n of Parameters by the Method of Moments 4"3 2-4. Maximum L i k e l i h o o d Estimators . . . 46 2-5. Covariance M a t r i x of Maximum L i k e l i h o o d Estimators A. Method of C a l c u l a t i o n . . . . '.. 52 B. C a l c u l a t i o n of the Elements of J * . 57 2-6. E f f i c i e n c y of Method of Moments A. Method of C a l c u l a t i o n . . . . 6 1 B. C a l c u l a t i o n of 'det M' i n Terms of the Parameters . . . . . . 62 C. Determination of the Jacobian 'det (}*' 64 1-7. Sample Zero Frequency and U n i t Sample Frequency Estimators A. Sample Zero Frequency and F i r s t Moments 66 B. U n i t Sample Frequency Estimators 68 v l page CHAPTER I I I L i m i t i n g D i s t r i b u t i o n s of the Poiss o n - M u l t i n o m i a l and Po i s s o n - Negative M u l t i n o m i a l D i s t r i b u t i o n s 3-1. I n t r o d u c t i o n 70 3-2. The Poisson-Poisson D i s t r i b u t i o n . . 70 3-3- The Information M a t r i x 73 3-4. E f f i c i e n c y of Method of Moments . . 75 3-5. Sample Zero Frequency and F i r s t Moments 78 3-6. U n i t Sample Frequency E s t i m a t i o n . . 79 APPENDIX 1A Obtaining an E x p l i c i t E xpression f o r the P r o b a b i l i t y F u n c t i o n . . 82 APPENDIX IB C a l c u l a t i o n of Moments from F a c t o r i a l Cumulant Generating Function . . 86 APPENDIX 1C C a l c u l a t i o n of the E n t r i e s of the Information M a t r i x " J " . . . 92 APPENDIX ID C a l c u l a t i o n of the Inverse of the M a t r i x J/£ 95 APPENDIX I E A Lemma Showing fl = CfTM0) . . . 100 APPENDIX I F C a l c u l a t i o n of (1-6.12) and (2-6.5) 102 APPENDIX 1G C a l c u l a t i o n of (1-6.16) and (2-6.6) 103 v i i page APPENDIX 1H C a l c u l a t i o n of "det M" . . . . 105 APPENDIX 2A Obtaining the P r o b a b i l i t y Generating Function g*(s) 106 APPENDIX 2B C a l c u l a t i o n of the E n t r i e s of the Information M a t r i x " J * " . . . 107 ) APPENDIX 2C C a l c u l a t i o n of det (}* . . . . HQ APPENDIX 3A C a l c u l a t i o n of E f f i c i e n c y of Method of Moments f o r the P o i s s o n - M u l t i - v a r i a t e P o i s s o n D i s t r i b u t i o n . . 112 BIBLIOGRAPHY 115 i i v i i i ACKNOWLEDGMENT i The author would l i k e to express h i s sincere thanks to Dr. Stanley w. Nash whose many hours of guidance and constructive c r i t i c i s m made t h i s thesis possible. He would also l i k e to thank Dr. R.A. Restrepo f o r reading the f i n a l form, and Miss Carol Lambert f o r typing i t . F i n a l l y , he wishes to thank the National Research Council of Canada and the Un i v e r s i t y of B r i t i s h Columbia for t h e i r f i n a n c i a l assistance. 1 INTRODUCTION I n recent years there have been many attempts t o i n v e s t i g a t e s t a t i s t i c a l l y the behavior of va r i o u s i n s e c t and p l a n t p o p u l a t i o n s . I t has been found t h a t models u s i n g a simple normal, P o i s s o n , or bi n o m i a l d i s t r i b u t i o n are g e n e r a l l y inadequate. The negative b i n o m i a l d i s t r i b u t i o n has been used somewhat s u c c e s s f u l l y by F i s h e r [1Q41], Anscombe [1950], B l i s s [1953], and o t h e r s . More r e c e n t l y , compound o r 'contagious' d i s t r i b u t i o n s have been a p p l i e d t o b i o l o g i c a l models w i t h somewhat g r e a t e r success. The three which are most commonly used are ( 1 ) , the P o i s s o n - b i n o m i a l - McGuire, B r i n d l e y , and Bancroft [1957], S p r o t t [1958], Shumway and Gurland [ i 9 6 0 ] ; (2) the Poisson-Pascal (or Poisson- negative b i n o m i a l ) - K a t t i and Gurland [ 1 9 6 l ] | and (3) the Poi s s o n - Poisson (or Neyman Type A) - Neyman [1938], Douglas [1955]. Models based on these d i s t r i b u t i o n s , however, must assume homo- geneity i n the c h a r a c t e r i s t i c s of the experimental p l o t . These might i n c l u d e s o i l type, amount of moisture present, type of ve g e t a t i o n p r e s e n t , e t c . Attempts t o r e l a x t h i s assumption or to g e n e r a l i z e i n other ways have been made by Neyman [1938], F e l l e r [19^3], Thomas [19^9], B e a l l and R e s c i a [1953], and Gurland [1958]. So f a r , o n l y the u n i v a r i a t e case has been considered f o r the above compound d i s t r i b u t i o n s and t h e i r g e n e r a l i z a t i o n s . The object of t h i s t r e a t i s e i s t o extend some of the r e s u l t s of the three compound d i s t r i b u t i o n s mentioned above t o the m u l t i v a r i a t e case. 2 CHAPTER I THE POISSON-MULTINOMIAL DISTRIBUTION 1-1. A B i o l o g i c a l Model I n t h i s model we assume there i s a l a r g e f i e l d A of area S^, and homogeneous throughout, where batches of i n s e c t eggs are l a i d . Homogeneity i m p l i e s t h a t the p r o b a b i l i t y d e n s i t y of the batches f o l l o w s a uniform d i s t r i b u t i o n over the f i e l d . We w i l l a l s o assume the p o s i t i o n of a p a r t i c u l a r batch i s independent of the p o s i t i o n s of the o t h e r s . T h i s seems t o be a reasonable assumption as l o n g as the average d i s t a n c e between the batches i s much g r e a t e r than t h e i r s i z e . Next, l e t us choose a r e g i o n B of A which i s f a r enough from the boundary of A so t h a t boundary e f f e c t s w i l l be n e g l i g a b l e . L e t us d i v i d e B i n t o many small p l o t s or quadrats, B^, Bg,' , a l l having the same shape and area Sg , which i s much smaller than S^. L e t Z be a random v a r i a b l e denoting the number of batches l a i d i n a p a r t i c u l a r quadrat. I f M i s the t o t a l number of eggs l a i d i n A, then p B o l z 1 a ° SA \ J Mz I f M i s assumed to be l a r g e , since Sg << S A, t h i s i s a p p r o x i - o mately the Poisson d i s t r i b u t i o n P(Z = z) = e" X - X where X = M z l . "Ho SA (1-1.1) Having outlined the breeding ground, l e t us consider the insects themselves. We suppose the insects coming from those eggs that' hatch can be divided i n t o n-1 classes on the basis of some dis t i n g u i s h i n g c h a r a c t e r i s t i c (e.g. colour, s i z e , type of insect, etc. ). For each integer i , 1 _< i jC n-1, l e t be a random variable denoting the number of insects i n a quadrat that are born i n t o the i c l a s s . We assume the p r o b a b i l i t y of an insect being born i n t o the i class i s and i s independent of what happens to any other i n s e c t . The p r o b a b i l i t y p n that an insect does not hatch i s therefore n-1 i = l n - 1 " I P i 1 " 1 - 2 ) We now make the a r b i t r a r y assumption that exactly N eggs are l a i d i n each batch. Assume a l l the eggs hatch about the same time, and sometime l a t e r we count the number of insects i n a quadrat, noting how many belong to each c l a s s . I f we assume the e f f e c t of insects migrating into and out of the quadrat i s negligable, we have that the conditional j o i n t density of X^, X 2, • • • > -^n-l """S ( $\z*z) = P^ (x = xjz - z) = (|z) T T P ± X I (1-1.3) 4 where we def i n e X - (X^, Xg, ) I J- * n i = l 0 otherwise n-1 where x n = Kz - ̂  x^ i = l 0 (1-1.4) J Thus we have a m u l t i n o m i a l d i s t r i b u t i o n . We can combine (1-1.1) and ( l - l . J ) t o get the j o i n t d e n s i t y P(X=x) = P-(x) = £ P(X=x | Z=z) P(Z=z) z=o n .-X 1$ d2)TTp x i (1-1.5) z=o i = l 1-2. P r o b a b i l i t y Generating Function and Recursion Formula f o r P r o b a b i l i t i e s G e n e r a l l y i t i s much e a s i e r t o c a l c u l a t e i n d i v i d u a l p r o b a b i l i t i e s u s i n g a r e c u r s i o n formula r a t h e r than the d e n s i t y f u n c t i o n . The f i r s t step i n t h i s d i r e c t i o n i s t o f i n d the prob- a b i l i t y generating f u n c t i o n g ( s ) where t = ( s ^ s^9 — , s n - l ^ g(S) = B ( S l X l s / 2 ... s ^ * " " 1 ) z=o x n =o x_ -. =o i = l 5 We may assume the upper l i m i t of the sums t o he » "because of the d e f i n i t i o n of (|z) given i n (1-1.4). To s i m p l i f y the above expr e s s i o n , we simply note t h a t the n-1 summations on the r i g h t are the m u l t i n o m i a l expansion of n ^ Mz 1=1 Then » z n-1 z=o z ' i = l and the r e s u l t of summing t h i s i s n-1 g ( s ) = e x p | x [ ^ s ± v ± + p n ] N - x | (1-2.1) 1=1 j Prom (1-2.1) and the d e f i n i t i o n of a p r o b a b i l i t y generating f u n c t i o n , we can c a l c u l a t e the i n d i v i d u a l p r o b a b i l i t i e s by means of n-1 _ (1-2.2) 1=1 •4* V* where i s the (n-1 )-vector w i t h 1 i n the k p o s i t i o n and zeros elsewhere, and means the p a r t i a l d e r i v a t i v e w i t h respect t o . Using the L e i b n i t z r u l e f o r m u l t i p l e d i f f e r e n t i a t i o n , we o b t a i n the f o l l o w i n g r e s u l t s which are e x p l i c i t e l y c a l c u l a t e d i n Appendix 1A. 6 P^(x +e f e) = P x ( 0 ) = g(6) = exp [ X ( p n H - l ) l (1-2.3) 1=1 n-1 p iJTT (-) 1=1 1 ] P*(y) 1-3. E s t i m a t i o n of Parameters by the Method of Moments The f i r s t step i n t h i s method i s n e c e s s a r i l y t o f i n d the moments of the d i s t r i b u t i o n . T h i s can probably be best accomp- l i s h e d i f we r e a l i z e t h a t g ( s ) i s the f a c t o r i a l moment generating f u n c t i o n i f we set s = (1, 1, , 1) s i i n s t e a d of s=0. This i s the f a c t o r i a l cumulant generating f u n c t i o n from which we can e a s i l y c a l c u l a t e the f a c t o r i a l cumulants. Then, u s i n g t a b l e s r e l a t i n g moments and cumulants such as the one i n David and Barton [1962], pages 142-3, we may f i n d the f a c t o r i a l moments and f i n a l l y the moments about the o r i g i n . Since the above mentioned t a b l e r e l a t e s cumulants and moments about the o r i g i n , i t a l s o r e l a t e s f a c t o r i a l moments and cumulanbs since both have the same r e l a t i o n s h i p s . (David and Barton [1962], page 51). Now set c ( s ) = l o g g ( s ) = X j [ Y s t P i + P n l W - l i = l (1-3.1) 7 Using the above procedure, we f i r s t d e f i n e G, = N(X+1) - 1 Grt = G, = N 2(X 2+3X+1) - 5N(X+1) + 2 N5(x3+6x+7X+l) - 6N2(X2+3X+1) . + 11N(\+1) - 6 Then the moments are, according t o Appendix IB, E(X ±) = NXP ± E ( X ± 2 ) = NXp i(p iG 1 +l) = NXp^PjNCx+l) - p±+l] E f X ^ j ) = NXp 1P JG 1 - NXP^jtNCX+l) - 1] E ( X i 5 ) = Nxp1(pi2G2+3PiG1+l) E ( X i 2 X j ) . HXP 1P J(P 1Q 2+G 1) E(\Xj\) = N X P l P d P k G 2 E ( X ± 4 ) = lXP i(p i 3G 3+6p i 2G 2+7p iG 1+l) E i X ^ X j ) = NXp 1P J(p i 2G 3 43p iG 2+G 1) E ( X 1 2 X J 2 ) = Nxp iPj[p 1P JG 3+(p i+p J)G 2+G 1] = K ^ i * W m G 3 E f X ^ X . ^ ) = Nxp iPjP k(p iG 5+G 2) (1-3.2) (1-5.3) (1-3.4) Only the moments given i n (1-3.3) are needed t o estimate the 8 parameters. The remaining ones, however, w i l l be needed l a t e r t o c a l c u l a t e the e f f i c i e n c y . L e t us def i n e a new random v a r i a b l e ¥, n-1 ¥ = Y X, (1-3.5) i = l 1 n-1 Then E(¥) = £ E ( X ± ) = N X ( l - p n ) (1-3-6) i = l n-1 n - l and E(W2') = E( V V X.X.) 1=1 j = l 1 3 n-1 n-1 -I E ( X I 2 ) + I Y E < x i V i = l 1=1 J4I I f we s u b s t i t u t e f o r the expectations from (1-3.3) and sum, E(W 2) = N X ( l - P n ) [ l + [ H ( X + l ) - l ] ( l - P n ) J (1-3.7) S u b s t i t u t i n g (1-3.6) i n t o (1 -3.7i™ e o b t a i n E(¥ 2) = E(¥) [E(¥) + ( N - l ) ( l - P n ) + l ] (1-3.8) Now we can solve (1-3.6) f o r 1-Pn> s u b s t i t u t e i n t o (1-3.8)> and solve the r e s u l t i n g equation f o r X. ¥e o b t a i n ,2j X = I z l (1.3.9) N E(W 2)-E 2(¥)-E(¥) From (1-3.3) P i - E ( X ± ) E(X j L)[E(W 2)-E 2(W)-E(W)] NX (N-1) E*(w) n-1 (1-3.10) i = l * K (1-3-9) and (1-3.10) give the moment estimators X* and p^* f o r the paramters X and p^ r e s p e c t i v e l y , i f the. p o p u l a t i o n moments are rep l a c e d by the corresponding sample moments. Before proceeding, l e t us de f i n e some n o t a t i o n . L e t p be the number of samples we observe. Next d e f i n e where X j Q i s the obs e r v a t i o n of the J** 1 c h a r a c t e r i s t i c from the a t h sample. F i n a l l y , d e f i n e P n-1 0 j = i j = i a=i n-1 w = ) x. a L ja 3=1 ( l n 3 . l l ) t h So, i s the mean of the number of i n s e c t s w i t h the i c h a r a c t e r i s t i c per sample and x.. i s the mean number of i n s e c t s per sample. Thus, from (1-3.9), we may w r i t e 2 r - i 3C« • x* = T H (1/0) Y w a 2-x.. 2-x.. a=l 10 A l s o , p^^ i s estimated by observed no. of i n s e c t s w i t h i property no. of i n s e c t s observed + estimated no. of unhatched eggs i . e . p ± * = x^/NX* and p n i s estimated by estimated no. of unhatched eggs no. of i n s e c t s observed + estimated no. of unhatched eggs n-1 p n * = 1 - Y P i * = (Nx-* " x..)/NX* i = l 1-4. Maximum L i k e l i h o o d E s t imators The l i k e l i h o o d f u n c t i o n , L, i s given by L =TT Px^J (1-4.1) a=l To f i n d the maximum l i k e l i h o o d e stimators \ and p^ of X and p^, we must solve the f o l l o w i n g system of equations. dL/aX = 0 (1-4.2) S L / ^ = 0 1 = 1, 2, n-1 Equation (1-4.1) can be w r i t t e n as f o l l o w s l o g t - J l o g ( X Q ) a=l 11 Because " l o g L" i s a mcnotone i n c r e a s i n g f u n c t i o n of L, (1-4.2) i s e q u i v a l e n t t o s o l v i n g the system 3 — l o g L = y i — 1- P_,(£ ) = 0 o (1-4.3) l o g L = T i 2- P-^x* ) = 0 Using (1-1.5) we can see t h a t - a - P X ( K) = e-^ d ^ f f p ^ 1 £ - ̂ ] * J Z = o z ! 1=1 Pn Let us expand t h i s expression i n t o two terms. I n the second term r e p l a c e ( S z ) by C j J ^ H x j + 1 ) since these two expressions are equal. Then i t i s c l e a r t h a t x f ... _ a r~i i ̂  . M"» P< .-rV x<. x-r^ 1 an X n X u 7.» X + e 1 n 1 , 1 , -n (1-4.4) P j z i o 2 1 J Pn i = l ' P j X, x.+l = -1 P->(x) - - i — P^x+e,) P j P j * ' Equation (1-4.4) must ho l d f o r each o b s e r v a t i o n , i . e . i t holds f o r x=x a and x j = x j o ; • S u b s t i t u t i n g t h i s i n t o (1-4.3) we o b t a i n it l o g L = £ p-TTT c? W-^p*<W ]-° B P j a=l P j , P j M u l t i p l y i n g by p^ and u s i n g (1-3.11), t h i s becomes 12 . £, f x , +1 )P^(x +e.) $ _L. i o g L = px, - 7 { 3a x K a J = o (1-4.5) S P j J ' ak P ^ ( x a ) Considering again the p r o b a b i l i t y f u n c t i o n (1-115), we can d i f f e r e n t i a t e w i t h respect t o X and o b t a i n oo „ n n-1 9A- z=o z* i = l Equation (1-1.4) i m p l i e s z = x n/N + (1/N) £ x k . k=l I f we s u b s t i t u t e f o r the f i r s t z i n (1-4.6) u s i n g t h i s expression, then SX _~_ N z! x ' 1 1 z=o i = i + * - x i i ("r " k ) ^ ? d ^ f f p ^ 1 - ^ ) z=o k=l ' i = l P m N X z=o 2 1 m P n i = l N X k=l where m may be any i n t e g e r between 1 and n-1 i n c l u s i v e . Thus n-1 9X - p mNX - - N X k = 1 l _ P _ , ( x ) = P n ( V " 1 ) P x(x+r m ) + (— £ x k - l ) P x(5c) (1-4.7) We should n o t i c e at t h i s p o i n t t h a t we a c t u a l l y have n-1 expres- sions f o r 3/3X(P^(x)). These are a l l equal, however, and i t 13 does not matter which one we choose. Equation (1-4.7) holds f o r each o b s e r v a t i o n , i . e . when x = x a , a x ^ a . Then, s u b s t i t u t i n g (1-4.7) w i t h t h i s m o d i f i - c a t i o n i n t o the second equation of (1-4.3), we o b t a i n I P«(W + I T , . !] = 0 (1-4.8) M u l t i p l y i n g t h i s by i t p m , s u b s t i t u t i n g from (1-4.5) f o r the f i r s t term, and u s i n g the s i m p l i f y i n g n o t a t i o n d e f i n e d i n (1-3-11), I f we sum over m, t h i s y i e l d s x.. - N X ( l - P n ) = 0 Hence P n = (NX - x.. ) / NX (1-4.10) F i n a l l y , l e t us s u b s t i t u t e t h i s i n t o (1-4.9) P m = / NX m = 1, n-1 (1-4.11) Equations (1-4.1Q) and (1-4.11) give the maximum l i k e l i h o o d e s t i m a t o r s f o r the p^ i f we know the corresponding estimator f o r X. We could conceivably solve f o r % by s u b s t i t u t i n g (1-4.10) and (1-4.11) i n t o (1-4.5) or (1-4.7) w i t h the expression set equal t o zero. However, i t i s not hard t o see t h a t i t would be impo s s i b l e t o solve t h i s d i r e c t l y f o r X. Thus i t i s necessary t o use a numerical procedure. 14 The following c a l c u l a t i o n i s "based on Newton's method, which says that i f ffx) = 0, then where X^ i s the n t n i t e r a t i o n of X and /\ Q i s the i n i t i a l estimate, which might be the moment estimator of X. To f i n d a suitable f("5i) i n our case, note from (1-4.9) that V&m - (** - (1-^.13) 6 n-1 I f we substitute f o r P n/^ m and £ £ x k a i n (1-4. 8), using a=l'k=l (1-4.13) and (1-3-11) respectively, and multiply the r e s u l t i n g equation by NX/(N"^-x..), we obtain an expression which i s zero. We may use t h i s as our f ( l ) . • Hence 5, (x +1) P-?(x „+e m) f ( t ) = Y ! - P - 0 (1-4.14) a=l xm. P x ^ a ) I t remains to f i n d D-^f(x) f o r substitution i n t o (1-4.12), From (1-4.14), 6 *__+! a=l "'Sa. n-1 3 5 2 ( X « J (1-4.15) Now D ^ x J . £ J j - P ^ x J - D # k P 4 (2 a ) (1-^.16) k=l 9 P k d * and from (1-4.11), D^pm = - x m / N l 2 = - p f f l A (1-4.17) Let us substitute expressions f o r the derivatives i n (1-4.16). 15 Use (1-4.4) t o replac e 9 / a p k [ P - ( x a ) ] , (1-4.7) t o replace d/dXtP^x* ) j and (1-4.17) t o replace D<?p . Then we may use AO, A. m (1-4.13) t o replace f n / P m and (1-3-H) t o replac e the r e s u l t i n g e xpression. n-1 D * - <V*) I ( W 1 ) " K(N-Dw a/NX> + 1 ] k=l n\ * (1-4.18) in. w x I f we replace x\ by x +e . and hence x by x. +1, and a " , a m' ma ma n-j use (1-3.11) t o replace y x ka k=l n-1 k=l - [ 1 + ( w a + l ) ] P x ( x a + ^ m ) + ^ + N % ' X ' - 1 ' ^ 3 NX x a m ^ + 1 ^ • ^ ( V 2 ^ ) (1-4.19) The (r+1) i t e r a t e d values of X, p^, P n_]_ C a n be c a l c u l a t e d from the r i t e r a t e d values by the f o l l o w i n g procedure. F i r s t s u b s t i t u t e the r t n i t e r a t e d values i n t o (1-4.18) and (1-4.19) t o o b t a i n D*P--.(xV) and D«P^(x +eL). Then s u b s t i t u t e these X x x a ' X x v a m' i n t o (1-4.15) t o f i n d D ^ f ( t ) which i s f i n a l l y s u b s t i t u t e d i n t o (1-4.12) f o r X r + 1 - Then the ( r + l ) s t i t e r a t e d values of the p m can be found from (1-4.10) and (1-4.11). 16 1-5. Cover!ance M a t r i x of Maximum L i k e l i h o o d Estimators A. Method of C a l c u l a t i o n /\ D i r e c t c a l c u l a t i o n of the covariance m a t r i x , fl , of the maximum l i k e l i h o o d estimators i s p r a c t i c a l l y i m p o s s i b l e . Under c e r t a i n c o n d i t i o n s , however, we may f i n d the asymptotic fl as P ->• * by f i r s t c a l c u l a t i n g the i n f o r m a t i o n m a t r i x J = XX I xp 1 * * * I * p n _ 1 I , I P XX P 1 P 1 O 0 o I, PnP l ^ n - l I p n - l x I p n - l X I p n - l p n - l : (1-5.1) where l f l t = E ( | - l o g L • l o g L ) = - E ( ^ l o g L ) (1-5.2) where L i s the l i k e l i h o o d f u n c t i o n . The second e q u a l i t y i s tr u e by the argument presented i n K e n d a l l and St u a r t [1961] PP. 52-53. Prom the remarks at the beginning of cj 1-3 we conclude t h a t the f a c t o r i a l moment generating f u n c t i o n i s gi v e n by (1-2.1) where s* i s set equal t o 1 Instead of 0. Since g ( s ) i s c l e a r l y i n f i n i t e l y d i f f e r e n t i a b l e , a l l the f a c t o r i a l moments are f i n i t e . Because each moment about the o r i g i n i s a f i n i t e l i n e a r combination of f a c t o r i a l moments, these moments are a l s o f i n i t e . Lemma 1-1. Let X,Y, and Z be non-negative, i d e n t i c a l l y d i s t r i b u t e d 17 and mutually independent random v a r i a b l e s w i t h E(X^) < ». Then 0 < E(XYZ) < EiX2!) < E ( X 5 ) . Proof; Consider 0 < E(XY^ - Y^Z)2 = E(X 2Y) - 2E(XYZ) + E ( X Y 2 ) . Because of the mutual independence and i d e n t i c a l d i s t r i b u t i o n of the random v a r i a b l e s , E f X 2 ! ) = E ( X 2 ) E(Y) = E ( Z 2 ) E(Y) = E ( Y Z 2 ) . Thus 0 < E(X 2Y) - E(XYZ). Now, n o t i n g t h a t XYZ > 0 always, we have 0 _< E(XYZ) _< E(X 2Y) Now consider 0 < E ( X 5 / 2 - X ^ ^ l i f ^ E ( X 3 ) - 2E(X 2Y) + E ( X Y 2 ) . But E (X 2Y) a E(XY 2) since d i s t r i b u t i o n s are i d e n t i c a l . There- f o r e 0 < E ( X 5 ) - EiX2!) and so E(X 2Y) < E ( X 5 ) . Combining t h i s w i t h the previous r e s u l t we have 0 < E(XYZ) < E(X2Y) < E ( X 5 ) . Q.E.D. Lemma 1-2. For the Poisson-multinomial d i s t r i b u t i o n n-1 n - l P x ^ + % i ) < (Pn/Pn>( I X i + N + X N ( I x i + N ) N } P x ( x ) <• 1=1 i = l 3 Proof; From (1-1.5), P x ( x + r m ) . e" x J ( x V * D (gw ) ( T T P i X i K P m / P n ) d " " ) z=o m 1=1 Let 1T be an i n t e g e r such t h a t 18 n-1 N ( T - 1 ) < £ x± < NT (1-5.4) 1=1 T depends on x. Prom the d e f i n i t i o n of the mu l t i n o m i a l coef- f i c i e n t , we see tha t the f i r s t T terms of the sum i n (1-5.3) are zero. Thus, i f we w r i t e the mul t i n o m i a l c o e f f i c i e n t i n a s l i g h t l y d i f f e r e n t way, 0 0 z „ _JL x. Nz-x -...-x.. >*<**w> - I 77 (iz> ( T T P i 1 ) ( p m / p n ) z=T z ' i = l V*" 1 oo n < (P„/P N) I (xz/z:) (f) (TTPI ) " z=T n i = l n » - ( P m / P n ) { m ( x T / T 0 ( f ) ( T T P i " 1 ) * • tj^JT (f) (TT P ^ 1 ) ) 1 = 1 z=T+l 1 = 1 (1-5.5) Note t h a t the f i r s t term i n the brackets i s NT times the 'z=T" term i n the expansion of P-j(x) and, since each term i s p o s i t i v e , n n-1 NT(X T/T!)(| r) ( T T P i i ) < NTP^x) < ( £ Xj4-N) P-(S) (1-5-6) 1 = 1 1 = 1 n-1 Now consider the second term i n (1-5.5). Because £ x i < NT 1=1 and z 2 T + l * ro* (Nz-N)! 5 l i (Nz-k) ( | Z ) = T ^ H ^—5=1 JJ L-n=t (1-5.7) ( J ] " x ± l ) ( N z - N - £ x±)l k=o (Nz-k- £ x ± ) 1=1 1=1 1=1 Now since k<N and z-l>T we have 19 Nz-k- £ x ± Nz-N- 2, x ± NT- £ x ± 1=1 1=1 1=1 1=1 Thus (1-5.7) hecomes i = i Hence the second term on the right side of (1-5-5) i s l e s s than or equal to z=T+l^ z" x ;- „ , i = l 1=1 n - l This expression i s XN( ̂  x^+N)^ times part of the expansion i = l n-1 of P^(x) and thus i s l e s s than XN( £ x ±+N) N. i = l Using t h i s f a c t along with (1-5.6) i n (1-5-5) we have n-1 n-1 . P ^ x + e J < (P m / P n ) [ £ x i + N + X N ( £ x 1 + N ) N |p 3 e(x) i = l i = l Q.E.D. Lemma 1-3. For the Poisson-multinomial d i s t r i b u t i o n , E[io/oX(log L)P]<«P Proof i Using (1-4.1) and d i f f e r e n t i a t i n g the logarithm, E ( | i L l o g L p ) = E ( | I T j r l ^ l x P * ( ^ ) l 3 > OA i X Gt a=l P P P a/BX[P-(* a)] o/dXfP^x )] a/oXCP^Xg)] < } } ) e( X — 2 L * Y % " a = l Y = 1 6 = l W P x ( * V p*<*6) 20 Because the observations are mutually independent, the expressions o/SX[P^(x )] — are independent f o r a = l , 2, , f3. Hence we may apply lemma 1-1 t o the above i n e q u a l i t y . X a=l PX<X> Since the observations are i d e n t i c a l l y d i s t r i b u t e d , the above expression i s independent of a, hence E( \L. l o g L | 5 ) < ̂  E( | -2± p ) (1-5.8) ax " P^(x) and s u b s t i t u t i n g f o r a/aX[P^(x)] from (1-4.7), ^ , , p f x + l ) P^(x+eV) , E(|-L- l o g L p ) < (J* E ( | n ^ . x V _ A m + (1/NX) V x . - l p ) 1 9 X " % N X **<x> i = i Replacing the absolute value of the sum by the sum of the absolute values and u s i n g lemma 1-2, i Jr- £ x i + ( v 1 ) [ 1 A + < I! X i + W ) N ] +1) <• W A- i = l i = l When the above expression i s expanded, i t w i l l y i e l d a f i n i t e sum of terms of the f o l l o w i n g type - n-1 constant • E ( j [ x^ ̂ ) d=l ? (1-5.9) where the n^ are non-negative i n t e g e r s These terms are a l l f i n i t e s ince we know a l l the moments are f i n i t e Hence the r e s u l t f o l l o w s . Q.E.D. 21 Lemma 1-4. For the Poisson-multinomial d i s t r i b u t i o n E( 13/3 P ± ( l o g L ) P ) < oo , i = 1, 2, ..., n-1. Proof; By the same argument as i n lemma 1-3 but u s i n g i n s t e a d of X , we w i l l o b t a i n (1-5.3) w i t h p.̂  r e p l a c i n g X , i . e . E ( | ^ l o g L p ) < p 3 E ( , ^ i H x f £ l l p ) BPi P S ( x ) S u b s t i t u t i n g f o r the d e r i v a t i v e u s i n g (1-4.4), we f i n d * ^ x. x .+1 P-*(x+e.) , l o g L p ) < p 5 E ( U - J ^ i — J - P ) s p ± " p d p d P X(X) Replacing the absolute value of the sum by the sum of the absolute values and u s i n g lemma 1-2. < e ( * i - [ y x. + H + XN ( V" x, + T$f]\ ( P j p n i = l i = i > Upon expansion, the above expression becomes a f i n i t e sum of terms of the type described i n (1-5.9 )> and by the same reasoning as was used t h e r e , E[ Ja/ap.^ ( l o g L ) p ] i s f i n i t e . Q.E.D. Let us appeal t o theorem 2, page 282 i n Rao [1947]. This theorem says the f o l l o w i n g - Let ii be the covariance m a t r i x of the maximum l i k e l i h o o d e s t i m a t o r s Ôg, and L be the l i k e l i h o o d f u n c t i o n . 22 Then, i f E D a / d ^ ( l o g L ) | 2 + n ] < i = 1, 2, n f o r some n > 0, J " 1 " j£! ^ (1-5.10) where J i s the i n f o r m a t i o n m a t r i x and 0 i s the number of samples observed. Lemmas (1-3) and (1-4) show t h a t the Poisson-multinomial d i s t r i b u t i o n s a t i s f i e s the c o n d i t i o n s of Rao's theorem i f we choose n=l. Hence (1-5.10) h o l d s , and thus f o r samples of reasonable s i z e we can make the approximation J " 1 = A (1-5.11) B. C a l c u l a t i o n of the Elements of J Before proceeding l e t us f i r s t prove the f o l l o w i n g r e s u l t . Lemma 1-5- £ £ (x + l ) ( x +l)P^(x+e )P x(x+^,) Define A ± . = -1 + £ ... 2, — ~ — ~ — I T — *L- x l " v An-1 „=o x_ , - o ^  p i P j P x ( x ) (1-5-12) Then A i j = = A, say, f o r i n , i , J , k = 1 , 2, n-1. Proof; A i j ~ ̂ k = A i J ' A i m + A i m ' Amk C D W - l - l x, =o . =o 1 n-1 ( x d + l ) P x ( x - f e ; j ) ( x m + l ) P x ( g + e a ) P j Pm 00 + 23 xl=° ^ - i - o N 2 * 3 ? ^ * ) P j L P k R e c a l l now t h a t (1-4.7) i s tru e f o r a l l values of m from 1 t o n-1. T h i s i s p o s s i b l e only i f P . ( x + e ; ) = i ^ ^ ( x + e k ) . Using t h i s f a c t , the terms i n the brackets of the above equation are zero and hence the whole expression i s zero. Q.E.D. Let us consider 1 ^ . Prom (1-5.2) and (1-4.1) P I u = E [ ( * _ l o g L ) 2 ] = E([ I 1- l o g P ^ ( x a ) ] 2 ) 3X a = i d*- P P a=l Y=1 Because the observations are independent of each other and a l s o have i d e n t i c a l d i s t r i b u t i o n s , I - p E ( [ l - l o g P - ( x ) ] 2 ) + p ( p - l ) E 2 [ ^ - l o g P - ( x ) ] (1-5.13) K K 3X x 3X x At t h i s p o i n t l e t us observe t h a t E [ l - l o g P-(x)] = E [ - J ^ - 1- P^(x)] ax x P-(X) ax x 00 00 00 00 - I - I 77 p x ^ = 7 7 £ - I xl=° ^-1=° xl=° xn-l=° - l - ( l ) - 0 (1-5-14) ax 24 U L A P^(x) p m - x V m ' Thus (1-5.13) becomes, a f t e r d i f f e r e n t i a t i n g the l o g a r i t h m , S u b s t i t u t i n g f o r the d e r i v a t i v e from (1-4.7) and u s i n g the d e f i n i t i o n of e x p e c t a t i o n , x 1=o x n_ 1=o x v ' V *m n-1 -n 2 + [(1/NX) £ x^-1] P-(x) \ k=l J I f we now square, replace the r e s u l t i n g sums by the moments they represent and use (1-5.12) along w i t h lemma 1-5, n-1 i I x x = p n 2 (A+l) + (1/N 2X 2) B[( £ \ ) 2 ] + 1 n-1 " (2P n/Pm H x) E t V Z \ ' 1 ) ] + ( ^ m ^ 2 ) E( Xm> _ 1 k=l n-1 - (2/NX) B( £ x k ) k=l We can evaluate the various moments by expansion and the use of (1-3.3). A f t e r s i m p l i f y i n g the r e s u l t i n g e x p r e s s i o n , we o b t a i n (1/0) I u = p n 2 A + ( l - p n ) ( N X + Fp n-P n)/NX (1-5.16) By a s i m i l a r procedure the other e n t r i e s of the i n f o r m a t i o n m a t r i x may be c a l c u M e d . The r e s u l t s are 25 (VP) I XP. h i \ = " P n H X A + W p n + 1 " p n u r2. 2 (VP) Ip p = N V A + N a ( 1 / P i + 1 - N ) (VP) I N 2X 2A + N\(l-N) f o r i ^ j P i P d (1-5.17) The c a l c u l a t i o n s of the above r e s u l t s are done i n Appendix 1C. Let us def i n e B P P = ( 1 / P ) I P I P J J 4 i Thus + NX/p, = ( V P) I _ _ PP i P . ^ (1-5.18) (1-5.19) J = 3 I f we s u b s t i t u t e (1-5.18) and (1-5.19) i n t o (1-5-1), B XX B B XP XP B p p+NX/ P l- B XP B_ PP B XP B. PP B p p + H X / P n - l (1-5.20) By (1-5.11) the i n v e r s e of t h i s m a t r i x i s fl . I n Appendix ID the i n v e r s e of such a m a t r i x i s c a l c u l a t e d i n d e t a i l . By making the appropriate a s s o c i a t i o n of v a r i a b l e s we have n-1 det J = p n f B u + ( B u B p p - B X p 2 ) ( l - P n ) / N X J J\ (HVPj.) (1-5.21) L i = l 26 var \ = J KX + V l - P n ) p B xx^ + (BXXBPP"BXP ) ( 1- pn> B X P P i p ' ; : BXX N A +( BXX BPP - % Ki^)' •• 1 (B, - B.„ ) p.p. [ BXX H X + <BXXBPP ' B ^ ) ( l - P n ) ] H X ~ 1 RP± ^ BXX BPP- BXP 2 ) P i 2 , var p. = i [ — — — ± - « J * ®x [ BxxN X + ^ BXX BPP" BXP 2)(i-P n)"" (1-5-22) C o r o l l a r i e s 2.1 and 2.2 i n Rao [19^7] s t a t e t h a t i f the d i s t r i b u t i o n s a t i s f i e s lemmas 1-3 and 1-4, then the maximum l i k e l i h o o d e stimators are minimum variance estimators f o r l a r g e samples and i n terms of the g e n e r a l i z e d v a r i a n c e , det S~l , are a s y m p t o t i c a l l y e f f i c i e n t . 1-6. E f f i c i e n c y of the Method of Moments A. Method of C a l c u l a t i o n The e f f i c i e n c y of a method of parameter e s t i m a t i o n f o r a m u l t i v a r i a t e d i s t r i b u t i o n i s defined t o be E f f = d e t CM (1-6.1) det C where C i s the covariance m a t r i x of the estimators f o r the method i n question and C^ i s the covariance m a t r i x of the minimum variance e s t i m a t o r s . Because the Poisson-multinomial d i s t r i b u t i o n s a t i s f i e s 27 the c o n d i t i o n s of lemmas 1-3 and 1-4, c o r o l l a r y 2-2 i n Rao [1947] s t a t e s t h a t the maximum l i k e l i h o o d estimators have minimum variance, /~* Thus, i n our case, C M = JTi and C = _T1 , the covariance m a t r i x of the moment es t i m a t o r s . Hence E f f = ( d e t / I )/(det ft ) and by (1-5.11), E f f (1-6.2) det n • det J To c a l c u l a t e i l , we w i l l f i r s t f i n d the covariance m a t r i x , M, of the moment es t i m a t o r s . L e t us d e f i n e W and "X̂ as f o l l o w s p n-1 N (VP) I ( E \ a r a=l k=l P \ - ( V P) £ X i a a=l (1-6.3) where X k a i s the random v a r i a b l e denoting the number of i n s e c t s observed w i t h the k c h a r a c t e r i s t i c on the a o b s e r v a t i o n . W2 estimates E(W 2) and 1^ estimates E ( X 1 ) . By d e f i n i t i o n var W c o v f ? , ^ ) eov^W 2,^) ... cov(iP,Z Q_ 1) 3 cov(¥ var "X̂  covCx^/X^) ... c o v ( T 1 , X n _ 1 ) cov(¥ ,X 2) cov(!T 1,Y 2) var "X"2 .. c o v O ^ / X ^ ) cov(¥ 2,T n_ 1) c o v ( 7 1 , T n _ 1 ) c o v C X g , ^ ^ ). .. var T n _ 1 J ( l - 6 . t ) 28 By Appendix IE we can approximate det fl by det A « (det (}) 2 det M (1-6.5) where det (} i s the Jacobian 3[X, p-,, ..., p„ T ] det 0] = _ 1 S=l (1-6.6) ^ ( W 2 ) , E O ^ ) , E ^ ^ ) ] B. C a l c u l a t i o n of det M i n Terms of the Parameters F i r s t we must express the elements of M i n terms of P x , p 2 , ..., P n - 1 ^ X. Consider P var "3^ = var [(1/p) £ X ± a ] a=l Because the observations are independent and i d e n t i c a l l y d i s t r i - buted f o r each o b s e r v a t i o n , P var ^ m ( 1 / p 2 ) £ var X i a = (1/p) var X ± a = 1 (1-6.7) = (1/p) [ E ( X ± 2 ) - E 2 ( X ± ) ] I f we now r e p l a c e the expectations by the expressions given i n (1-3.3), we w i l l have var X± = (1/p) NX V± [ p ± ( N - l ) + 1] (1-6.8) Now consider cov (T^, X j ) I 4 3 • p P cov ( X ^ X , ) = cov [(1/p) £ X i a , (1/p) £ X J y ] a=l Y=1 P P . - (1 / P 2 ) E [ l x ± a - p E ( X i ) H lx^ - p E ( X j ) 3 i a«l Y=l ; 29 The l a s t e q u a l i t y i s t r u e since E ( X i a ) = E(X^) f o r a l l a, A l s o , since the observations are independent, p covOC^Tj) - (1/P 2) YE{[X±a ~ E( Xi> 3 [ Xja " E ( X j ) ] ] p a=l + ( V P 2 ) I I E^±a ~ E < X i ) 5 ~ E<V3 a=lY4° But E [ X ± a - E ( X ± ) ] = E ( X i a ) - E ( X i ) = 0. Hence, since'the X±a are i d e n t i c a l l y d i s t r i b u t e d i n a, we'can w r i t e c o v f X ^ Z j ) = (1/p) E [ [ X 1 - E ( X i ) ] [ X J - E ( X J ) ] ] (1-6.9) - (1/P) [ E ( X 1 X J ) ^ - E ( X ± ) E ( X j ) ] S u b s t i t u t i n g (1-3.3) i n t o the above equation we o b t a i n c o v C x ^ X j ) = (1/p) N(H-1)X p ± p j (1-6.10) Now consider cov(¥ ,X^). By the same reasoning as before __ f §. n-1 n-1 p -) cov(W 2,^) = cov (1/p) I XiaV* ^/P) I \a ( a=l i = l 3=1 k=l J n-1 n-1 n-1 n-1 - w*& 1 1 E W j ) - EL\) I" I E<xiV} l i = l 3=1 i = l 3=1 ^ - (1/P ) { E ( X k 5 ) + 2 £ E ( X k 2 X j ) + £ £ E ( X k X i X J ) î k J4k,i n-1 n-1 + I s(\\2) - *(\) [ I E(x±2) + I Z E ( x i x 3 ) ] ( i 4 k i = l 1=1 3 4 i J (1-6.11) 30 By s u b s t i t u t i n g (1-3.3) and (1-3.4) i n t o the above expression, we o b t a i n an equation i n terms of the parameters which u l t i m a t e l y reduces t o cov(W 2,'X k) = ( l / 0 ) H X P k | G 2 ( l - p n ) 2 + 3 0 ^ 1 ^ ) - NXCl-p^Ea^l-p^ + (1-6.12) The steps between (1-6.11) and (1-6.12) a r e - o u t l i n e d i n Appendix I F . For s i m p l i c i t y , l e t us de f i n e E1 = G 2 ( l - P n ) 2 + 3G 1(1-P n) ~ N \ ( l - p n ) [ G 1 ( l - p n ) + 1] + 1 (1-6.13) Then cov(W 2,X k) - (1/p) N X p ^ (1-6.14) F i n a l l y consider var Using again the arguments of i d e n t i t y and independence, we o b t a i n var W2 = var j (1/p) £ ( £ X i a ) 2 f n - i ( 1 = 1 } = (i/p) E ( I x±)4 - E 2 ( I x±y I i = l i = l , n-1 n-1 n-1 n-1 n-1 n-1 - <v*» I I I I B<WA> - c I Z w^ri ( i = l t1=l k=l T3=l i = l j = l n-1 ( i = l J 4 l k 4 i , j m4i,j,k n-1 n-1 i=i J4I k 4 i , j 1=1 j+i 31 n-1 n-1 n-1 1=1 j 4 i 1=1 1=1 n-1 \ + r r E w n ( i- 6- i5) How s u b s t i t u t e (1-3.3) and (1-3.4) i n t o the above equation and s i m p l i f y as i s done i n Appendix 1G. Then Var Tn£ = ( l / p j H X ( l - p n ) U 5 ( l - p n ) ^ + 6 G 2 ( l - p n r + 7 ( ^ ( 1 ^ ) + 1 - N X ( l - p n ) [ G 1 ( l - p n ) + l ] 2 ^ (1-6.16) For s i m p l i c i t y we may de f i n e H, 2 = G 5 ( l - p n ) 3 + 6 G 2 ( l - p n ) 2 + 7G x(l-p n) + 1 - Hx(l-P n)CG 1(l-p n) + l ] 2 (1-6.17) Then var W2 = (1/p )Nx(l-P n )Hg (1-6.18) Thus, s u b s t i t u t i n g (1-6.8), (1-6.10), (1-6.14), and (1-6.18) i n t o (1-6.4), HX(l-p n)H 2 NXp-Ĵ  ... ^P n-1 H1 Nxp^ NXP 1[p 1(N-l)+l] ... N(N-1 JXp-jP^ N X P 2 H 1 N(N-l)Xp1P2 N(N-1 )XP 2P n_ 1 * • • • • • • • • • • N X p n - l H l N(W-1 )Xp1Pn_1 ... 'HXP^CP^CN-D+I] (1-6.19) 4 32 The determinant of t h i s m a t r i x i s c a l c u l a t e d e x p l i c i t l y ^ i n Appendix 1H. We need only set N' = N-1 and R = NX i n the ma t r i x i n the appendix and we have (1-6.19). Then n-1 det M = ( N X / p ) n ( l - p n ) ( " [ " [ p 1)[H 2(N-Np n+p n)-H 1 2] 1=1 (1-6.20) C. Determination of the Jacobian, det (fr To evaluate det 4 as defined i n (1-6.6) we must evaluate the determinant of the f o l l o w i n g m a t r i x - oX oE(W 2) 3P 1 0E(W 2) 9 p n - l aE(w^) oX 9E(X X) 3P 9P n-1 9E(X 1 ) dp- 9 E(X 1) 3E(X 2) ax SPX s p n - i (1-6.21) To f i n d the above p a r t i a l d e r i v a t i v e s we appeal t o (1-3*9) and (1-3-10), n o t i n g t h a t n-1 E(W) = Y E ( X i ) ' i = l A f t e r d i f f e r e n t i a t i n g we o b t a i n 33 oX 0E(¥ 2) N-1 E(W)[2E(frr )-E(W)3 N [E(¥^)-E2(¥)-E(¥)]2 E(X.) (N-1 )E^(¥) SP. 3E(X j ) 9P ± SE(X i) E ^ ± \ E(W)-2E(¥ 2) N-1 ' E^(¥) E ( X ± ) E(W)-2E(¥ 2) N-1 E^(¥) i f E(¥ 2)-E 2(¥)-E(¥) (N-1 ) E 2 ( W ) I t w i l l be more convenient i f we are able t o express the e n t r i e s of (f i n terms of the parameters. To t h i s end we may- s u b s t i t u t e f o r the expectations i n the above set of equations u s i n g (1-3.3), (1-3.6), and (1-3-7). A f t e r s i m p l i f i c a t i o n we w i l l f i n d oX • 3E(¥ii) ' oX 3E(X, ) N ( N - l ) ( l - p n ) ' 2 [ p n + N ( X + l ) ( l - P n ) 3 - l N ( N - l ) ( l - p n ) 2 ap± aE(¥ y) oP ± BE(X J ) oP ± o E ^ ) p i N(N - l )x( l-P n)^ P i [ l - 2 [ p n + N ( X + l ) ( l . p n ) ] } N(N-1 )X(1-P n)' i f ±45 P i f l - 2 [ p n + N ( X + l ) ( l - P n ) ] ] 1 72. N(N-1 )X(1-P n) NX (1-6.22) 34 The s u b s t i t u t i o n of (1-6.22) i n t o (1-6.21) gives an e x p l i c i t e xpression f o r (J. I t s determinant may he found by m u l t i p l y i n g the f i r s t row by p ±/X and adding i t t o the ( i + l ) s t row f o r 1 = 1 , 2, . .., n-1. T h i s w i l l give an upper t r i a n g u l a r m a t r i x which may be expanded by the f i r s t column t o give det Cj = - _ 1 (1-6.23) (NX ) n - 1 N ( N - l ) ( l ^ p n r I f we now s u b s t i t u t e (1-6.20) and (1-6.23) i n t o (1-6.5), d e t a = N V i ) V - P n > V r V > P l At t h i s p o i n t we can f i n d the e f f i c i e n c y by s u b s t i t u t i n g the above equation along w i t h (1-5.21) i n t o (1-6.2). Hence N 2 ( N - l ) 2 ( l - p n ) 5 E f f = 5 5- (1-6.24) [ B U N X + ( 1 - P n ) ( B u B p p - B X p * ) ] [H 2(N-Np n +p n )-H^ ] 1-7- Sample Zero Frequency and U n i t Sample Frequency Estimators A. Sample Zero Frequency and F i r s t Moments Sample zero frequency e s t i m a t i o n i s u s e f u l i f the zero sample, ( i . e . X = (5), occurs q u i t e f r e q u e n t l y . From (1-2.3), i f we set X=0 P^(0) = e x p [ x ( p n N - l ) ] (1-7.1) L e t us de f i n e F ( a ) t o be the frequency w i t h which 35 X = a = ( a ^ , a 2 , a ^) occurs i n p observations. Consider the estimator F ( 0 ) f o r E^O) E [ | F ( 0 ) ] = ^ 0 ) = e x p [ X ( p n N - l ) ] (1-7.2) Hence (1/p) F(o) i s an unbiased estimator f o r EJ(0). We may o b t a i n the sample zero frequency estimators ~ and p^ f o r \ and p^ by u s i n g the moment estimators f o r the f i r s t moments given by (1-3.3) and (1-6.3) f f . together w i t h the estimator j u s t defined i n (1-7.2) t o o b t a i n the equations X. = NXp^ (1-7.3) (1/p) F ( 6) = e x p [ x ( p n N - l ) ] To solve f o r X, p^, l e t us f i r s t add the top equation of (1-7-3) f o r i = l , 2, — , n-1. n-1 N X ( l - P n ) = I ^ i = l S o l v i n g f o r p n and s u b s t i t u t i n g i n t o the bottom equation of (1-7.3), n-1 l o g M i = f [ ( l - - i V 7, f-1 ] (1-7.4) P ^ 1 = 1 We can use a numerical method t o f i n d x, and then from (1-7.2), P ± = \/NX (1-7.5) 36 B. U n i t Sample Frequency E s t i m a t i o n I f the u n i t samples ( i . e . X = e k , k = l , 23 n-1). occur" f a i r l y f r e q u e n t l y i t may he advantageous t o use t h i s e s t i m a t o r . From (1-2.1) we can see t h a t * ( e k ) = M ! 2 = [exp [ \ ( p / - l ) ] ) l P n M P ^ (1-7.6) s=0 Consider the estimator (1/p) F(®j c) f o r ^ s ^ ) ' We. n o t i c e 1 —* —* that E[j| F ( e k ) ] = ^(© k).- Hence the estimator i s unbiased. Thus we may solve the equations (1/P) F ( e k ) = { e x p [ X ( p n N - l ) ) } N p n N _ 1 p k X k = l , 2, n-1 (1-7.7) along w i t h (1-7.3) f o r p^ and X t o o b t a i n t h e i r u n i t sample estimators p^ and X. To solve these equations l e t us d i v i d e (1-7.7) by (1-7.7) w i t h k = l . Then m * ( 1 . 7 . 8 ) P ( e x ) P X D i v i d i n g (1-7-7) w i t h k=l by (1-7-3) n ~ l v N-1 Fie^/FiO) = N ( l - I p k ) p xX k=l I f we s u b s t i t u t e f o r p k from (1-7-8) and solve f o r X , v p 1NF(0) p. n r _̂  N-1 1/X = 1 . \ [1 - - i — I (1-7.9) F ( e x ) F ( ^ ) £ k 37 Prom (1-7.3) and the f a c t t h a t the sum of the i s one, we o b t a i n n-1 e k=l Upon s u b s t i t u t i o n f o r p k from (1-7-8) and d i v i s i o n by X, we f i n d 1 l o g I l 9 l + i . [ i i — V P ( e k ) ] (1-7-10) X B F ( e ] L ) ^ I f we now s u b s t i t u t e f o r 1/H from (1-7-9), equation (1-7-10) becomes p,NF(0) p, _ N-1 w ? r x 1 [1 - — i — V P ( e t ) ] l o g H P J . + i P i V * = t l " p 7 h £ F ( * * ) 3 F ( e l ' k=l T h i s may be r e w r i t t e n as ^ n-1 , , )* ^ n-1 P l - - i _ £ P ( e k ) ^ f - ^ r r CHP(O) l o g + I F ( e k ) ] - l | + 1 = 0 (1-7.11) We can now use a numerical method t o f i n d p^. We can then c a l c u l a t e X from (1-7-9) and f i n a l l y p^, 1=2s 3, ..., n-1 from (1-7.8). 38 CHAPTER I I THE POISSON-NEGATIVE MULTINOMIAL DISTRIBUTION 2-1. A B i o l o g i c a l Model This d i s t r i b u t i o n a r i s e s from a model very s i m i l a r t o the one given i n § 1.1 f o r the Poisson-multinomial d i s t r i b u t i o n . The only v a r i a t i o n s are the f o l l o w i n g . L e t N represent, i n s t e a d of the t o t a l number of eggs l a i d i n each batch, the mean number of eggs t h a t do not hatch i n each batch. Let Z be a random v a r i a b l e denoting the number of batches of eggs l a i d i n a p a r t i c u l a r quadrat and assume the egg l a y i n g stops as soon as the (Nz) egg i s l a i d t h a t w i l l not hatch. Hence i f we d e f i n e p 1 ? p 2 , Pn_-j_ the same as i n §1.1 and p n by (1-1.2), then ( x 1 + . . .+x n_ 1+Nz-l)! x ± (2.1.1) •d rsw * \ i n - i ^ NZ-TT ^ P-(x|Z = z) = p n I | P i x 1 ! x 2 ! . . . x n _ 1 ! ( N z - l ) ! i = 1 i f z>0 P-(x|Z = 0) = ( 1 i f x=0 0 otherwise I f Z again has a Poisson (X) d i s t r i b u t i o n as i n § 1.1, then 00 P^(x) = £P x(*iZ = z ) P(Z=z) z=o (2-1.2) = e 2, ( x ' 1' — ; ~ p n 11 ?! 39 where we adopt the convention t h a t (Xj+.. .+^^+12-1)1 f l i f x = 0, z = 0 x ' .. .x„ , !(KTz-l)! \ 1 0 i f x 4 6, z = 0 (2-1.3) 2-2. P r o b a b i l i t y Generating Function and Recursion Formula f o r P r o b a b i l i t i e s The p r o b a b i l i t y generating f u n c t i o n i s defined by g*(s) = E ( S l X l s 2 * 2 ... B n_ 1* n- 1) (2-2.1) Thus from (2-1.2), (2-1.3), and (2-2.1) n-1 g* x.=o x , =o z=l x l ^ n - i ' V " 2 - 1 ^ ' i=l 1 n - l x,o ] + 6- where 6-± -> = | 1 i f x = 0 x,o 0 i f x 4 0 L e t us sum the terms i n 6^ se p a r a t e l y and rearrange the order of the sums of the other terms. Then g * ( s ) = £ . . . £ i T r ( s l P l ) z=l x 1 = o x n _ 1 = o x l x n - l * ^ z ± ) m ±=1 + e~ x (2-2.2) To evaulate the above expression we use the i d e n t i t y (x 1+...+x n_ 1+Nz-l): x l ! ' * ' ^ - I 1 ^ 2 " * 1 ) 1 n-1 n-1 / Z 3^+Nz-l^ X J 4o (2-2.3) Prom F e l l e r [1950], page 61, (12.4), we have the i d e n t i t y 03 00 Hence £ ("a) (-p) x = £ ( X + * r l ) ( - l ) X ( - P f = (1-p ) " a (2-2.5) x=o x=o The l a s t e q u a l i t y i s t r u e because the middle term i s simply the bi n o m i a l expansion of (1-p) • We may use (2-2.4) t o replace the c o m b i n a t o r i a l expression on the r i g h t side of (2-2.3) and then use the r e s u l t t o repla c e the f a c t o r i a l s i n (2-2.2). We o b t a i n an expression i n v o l v i n g negative b i n o m i a l c o e f f i c i e n t s z=l xn-l =° x i L = 0 3=1 n-1 / ~Y ^-Nz k=j+l (2-2.5i) n-1 • TT C-s±Pl f 1 + e i = l -X A f t e r c a r r y i n g out the i n d i c a t e d summations u s i n g (2-2.5) as i s done i n Appendix 2A, g z=o n-1 i = l -Nz 41 and hence N g*(s) = exp ) X *n •I 1=1 s,, P - X (2-2.6) L e t us consider the f o l l o w i n g change of v a r i a b l e s *n ' n p ± = - t o ± A n i = 1, . . . , n-1 N = - V > 0 (2-2.7) Then (2-2.6) becomes g*(s) = exp [ x [ b n + £ - \ n-1 i = l (2-2.8) Notice t h i s formula i s e x a c t l y the same as (1-2.1) where V corresponds t o N, and b t o p. Hence whatever we say about X and p^ i n the P o l s son-multinomial d i s t r i b u t i o n , we may say the same t h i n g about X and b^ r e s p e c t i v e l y i n the Poi s s o n - negative m u l t i n o m i a l d i s t r i b u t i o n by v i r t u e of (2-2.8). I n many r e s u l t s obtained i n t h i s chapter t h i s f a c t w i l l g r e a t l y reduce the l e n g t h of c a l c u l a t i o n s , while i n other s , e s p e c i a l l y those i n v o l v i n g d e r i v a t i v e s w i t h respect t o p^, i t i s b e t t e r t o c a l c u l a t e d i r e c t l y . To o b t a i n an expression f o r the p r o b a b i l i t i e s we must d i f f e r e n t i a t e g*(s*) an appropriate number of times. S t a r t i n g 42 w i t h (2-2.8) we can e x a c t l y f o l l o w the procedure i n Appendix 1A w i t h the obvious change i n symbols and o b t a i n (1A-4) which w i l l be w r i t t e n as n-1 P x ( x + e k ) = n-1 X V n V - l x l x n - l Y ... Y v(v-i) ... [v- Y (x±-y±)i i = l V 1 y x=o y ^ - o r -r-r b i x xi " y i i = l V P x ( y ) Prom (2-2.7) i t i s c l e a r t h a t (2-2.9) Upon s u b s t i t u t i o n f o r the b's and V i n the above equation from (2-2.7) and (2-2.9), we o b t a i n P*(x+eV) = - X ( P k / P n ) ( l / P n ) x k + 1 -N-1 x l x n - l Y ... Y (-BO(-N-l) .. • ^1=° yn-l=° ( x i - y i ) H T T ( - P i ) 1 \ x v ) t i = l i = l v x i - y i ; - ' 1 P*(y) I f we now f a c t o r the "minus one" out of each term immediately f o l l o w i n g the m u l t i p l e sum and each p^, and note t h a t N i s an i n t e g e r , we o b t a i n the second equation of (2-2.10). The f i r s t comes from (2-2.6). 43 P x ( x + e k ) = P^O) = g*(0) = e ^ n " 1 ) n _ 1 x P k p n N I1 V ^ h\-y±)v. y l = o y n - l = ° 1=1 (N-1)! (2-2.10) n-1 TT 1=1 T f P i X l " 7 1 / ( x i - y i ) ! ] P-(y) 2-3. E s t i m a t i o n of Parameters by the Method of Moments To o b t a i n the moments of t h i s d i s t r i b u t i o n , we use the same method as i n § 1-3. Prom (2-2.8) we may form the cumulant generating f u n c t i o n n-1 c ( s ) = x f [ b n + Y *±\f ~ l ] (2-3.1) 1=1 By f o l l o w i n g the c a l c u l a t i o n s of Appendix IB, but r e p l a c i n g N w i t h V, and p^ w i t h b^, 1 = 1, 2, n, we w i l l get (1-3.2), (1-3.3), and (1-3.4) w i t h the above replacement. L e t us c a l l these m o d i f i e d equations (1-3.2)', (1-3.3)', and (1-3.4)'. I f we apply the t r a n s f o r m a t i o n given by (2-2.8) and (2-2.9) so as t o express (1-3.2)', ( 1 - 3 . 3 ) % and (1-3.4)' i n terms of N and p ± and then d e f i n e G 1» = N(X+l) + 1 ^ 0 * = N 2(X 2+3X+1) + 3N(X+1) + 2 •x -z o I 3 o ; (2-3*2) P(X^+6\N-7X+1) + 6N^(X>3X+1) / G * = ^ ^ x +  ^(X +3X+1) + 11N(X+1) + 6 44 the moments w i l l he E(X±) = E ( X ± 2 ) = N\p ±/p n NXp, p, 1 (-± G * + 1) p n p n NXP4 p.H p. 1 ( X + 1 ) + - i + i] P, p n Pn n E(X±X^ = NXp ±Pj P. n NXp.p, A [N(X+1) + 1] n (2-3.3) E ( X ± 2 X J 2 ) E ( X i X / k ) NXp, p E ( X ± 5 ) = i ( i + G * + l ) p n P n n E ( X i X . ) . E ( X i X J X k ) - E ( X ± 4 ) NXp.p, p, g 3 (— G 2* + G x*) p n Pn ( G * 2 NXp, p n ^ 3 p 2 1 G *+6-L- G 9*+7— G *+l) Pn Pn' P n Pn 2 , 3 x NXP,p, p, p. E ( X i % j ) = — | l i (-^ G 3* + 3 G 2* + G 1*) n n n v n ^n p n 1 NXp, p.p. p. ±f-^ ( _ i G3* + G2*) Pn' P N x p i p j P k P i m (2-3.4) n 45 Let us now define the random v a r i a b l e , ¥, by n-1 W = £ X ± (2-3.5) i = l TiZ^ K \ ( l - p ^ ) Then E(W) = £ E ( X i ) = ~-^» (2-3-6) i = l p n n-1 n-1 and E(¥ 2) = E( £ £ X ±X^) i = l J=l n-1 n-1 = 1 E ( X ± 2 ) + £ I E ( X i X j ) i = l i = l 3 4 i I f we s u b s t i t u t e f o r the expectations from (2-3.2) and sum, E(¥ 2) = M ( l - p n ) ( 1 + [N(*+l)+l] ( 1 " P n ) | (2-3.7) P n I P n J Then, the s u b s t i t u t i o n of (2-3.6) i n t o t h i s y i e l d s E(¥ 2) = E(¥) JE(¥) + ( N + l ) ( l - p n ) + l j (2-3-8) " ~ P r T Consequently we can solve (2-3.6) f o r p n , s u b s t i t u t e i n t o (2-3.8), and solve f o r \ t o o b t a i n X . f i i » 2 ( V ) (2-J.9) N E(JT )-E*(W)-E(W) Prom (2-3.3), P i=P nE(X ± )/NX and from (2-3-6) pn=NX/[E(¥)+NX] Thus we may e l i m i n a t e p n from these two equations and s u b s t i t u t e f o r X from (2-3.9). A f t e r s i m p l i f y i n g 46 E(X±) E(Xi)[E(W2)-E2(¥)-E(W)3 ? i E(W)+NX E ( ¥) [E ( ¥ 2 )-E 2 ( W )+NE (¥) ] (2-3.10) P n = H\/[E(¥) + NX] To o b t a i n the moment estimators ' X* and p^* of X and r e s p e c t i v e l y , we simply replace the p o p u l a t i o n moments by t h e i r corresponding sample moments. Using the n o t a t i o n defined i n (1-3.11), (2-3-9) and (2-3.10) y i e l d x . . 2 H+l (1/3) £ w a 2-x.. 2-x.. a=l p i * = x ± / " [ x ' ' + n-1 p * = i - I p * = NX*/[x.. + NX*] n 1=1 1 2-4 Maximum L i k e l i h o o d Estimators I n t h l s s e c t i o n we w i l l see t h a t the d e r i v a t i o n of the maximum l i k e l i h o o d e stimators c l o s e l y p a r a l l e l s t h a t f o r the Poisson-multinomial d i s t r i b u t i o n . L e t us define 8, x^, and x ^ a as at the end of § 1.3. Then we may def i n e the l i k e l i h o o d . f u n c t i o n , L, as i n (1-4.1) and o b t a i n (1-4.3). For convenience we w i l l r e c o r d t h i s set of equations again 47 B log L = £ I _ A_ P-(x a) = 0 (2-4.1) a =1 P x ( x a > s p i Here, of course, denotes the Poisson-negative m u l t i n o m i a l r a t h e r than the Poisson-multinomial d e n s i t y . I t i s p o s s i b l e t o f i n d the d e r i v a t i v e s of P^(x) by d i f f e r e n t i a t i n g (2-1.2) a s , £ ,z (x, + ...+x„ ,+ H z - l ) l s p i zto 2 1 V-'.^.x^Nz-l)! n-1 'TTV° ( V p i - N z / p n ) 3=1 Consider the f o l l o w i n g i d e n t i t y n-1 n-1 z = (l/N)[(Wz + £ x j ) - £ X j ] (2-4.2) 3=1 3=1 I f we use t h i s i d e n t i t y t o s u b s t i t u t e f o r the l a s t z i n the above equation, then P-(x) - ( x . ^ ) P-(x) - e" x V n-1 x n-1 n-1 3=1 k=l k=l n-1 = [ x ^ + (1/P n)y x ^ P x ( x ) k=l 48 _ ( x i + 1 > e ~ X y £ {TV-...+xn_1+Vz)l p N z ^ n-1 ^ P i p n ' z=o z l x 1 : . . . x n _ 1 ! ( x i + l ) ( N z - l ) ! n ? i T T 3 =1 C l e a r l y t h i s reduces t o n-1 a - = + d/Pn) I ^ ] F S ( i ) - J £ P s ( X + ^ ) op^ K=X P n p i (2-4.3) How l e t us d i f f e r e n t i a t e (2-1.2) w i t h respect t o X. i . P.<*) = e" X y « £ i ( y - ^ i - i ^ ) ' p Nz 53 x d 9X X z ^ z! x 1 ! . . . x n _ 1 ! ( N z - l ) l n J i. ^ 3=1 " P x ( x ) The use of (2-4.2) t o s u b s t i t u t e f o r the f i r s t z i n the numerator of the above expression r e s u l t s i n = (e'VHx) I ^ [(Hz + £ x ± ) - £ x±]-± - f i l l „ _ z! i _ i i = i x.^ .. . x ^ ^ N z - l ) ' z=o 'n* IT Pj^ - 3=1 Therefore we s i m p l i f y t o W m j = 1 m=l, 2, n-1 (2-4.4) 49 Equations (2-4.3) and (2-4.4) w i l l h o l d f o r each o b s e r v a t i o n , i . e . when x = ~x a* ^ = xma, a = 1, 2, 6. With t h i s i n mind we may s u b s t i t u t e (2-4.4) i n t o the top equation of (2-4.1) t o o b t a i n V P*(y°m> . V t l + (I/Bx/V 1 - ] = o (2-4.5) Using the same i d e a we s u b s t i t u t e (2-4.3) i n t o the bottom equation of (2-4.1) a=l k=l a=l ^ ± I t i s p o s s i b l e t o replace the l a s t sum by s u b s t i t u t i n g from (2-4.5) and then s i m p l i f y i n g the n o t a t i o n by means of (1-3.11) to get x i y p i + x../p n - (Nx/£n)(B+x. ./NX) = 0. S o l v i n g f o r p^, t h i s becomes P ± = P n x i / l d i = 1, 2, n-1 (2-4.6) I f we now m u l t i p l y t h i s equation by Nx and add f o r 1=1, 2, n-1, we w i l l o b t a i n n-1 n-1 SIX P ± - F X ( l - P n ) = $n I * i . =^ nx.. i = l i = l Hence p n = NX/(NX + x..) (2-4.7) and upon s u b s t i t u t i o n of t h i s i n t o (2-4.6) we have the estimator f o r p ±. p ± = x± /(NX + x.. ) i = 1, 2, ..., n-1 (2-4.8) 50 I t s t i l l remains t o f i n d the estimator f o r X, tha t i s , X. As i s the ease w i t h the Poisson-multinomial d i s t r i b u t i o n , i t i s almost impossible t o solve f o r X d i r e c t l y , and hence we must use a numerical method. The f o l l o w i n g c a l c u l a t i o n i s based on Newton's formula which i s given by (1-4.12). W r i t i n g i t again, where t(t) = 0. (2-4.10) I f we use (2-4.9) t o s u b s t i t u t e f o r p m i n (2-4.5), the l a t t e r equation reduces t o an express i o n which i s a candidate f o r f ( X ) since i t s a t i s f i e s (1-4.13), i . e . The f i n a l step i n our procedure i s t o f i n d D ^ f ( l ) f o r s u b s t i t u t i o n i n t o (2-4.9). D i f f e r e n t i a t i n g (2-4.10) w i t h respect t o 1 gi v e s a=l *x v x a ' - P * < V V ^ P*<*a> ] (2-4.12) But we know m=l B pm B A l and from (2-4.7), Ik p. = - x, /(Nt + x . . ) 2 (2-4.14) A. J- 1 • 5 1 Now l e t us s u b s t i t u t e f o r the d e r i v a t i v e s i n ( 2 - 4 . 1 2 ) . Using ( 2 - 4 . 1 3 ) t o s u b s t i t u t e f o r the p a r t i a l s w i t h respect t o and 'i r e s p e c t i v e l y n - 1 n - 1 3=1 ^ i = l Next, s u b s t i t u t i n g f o r p n and p^ u s i n g ( 2 - 4 . 7 ) and ( 2 - 4 ^ 8 ^ r e s p e c t i v e l y , and u s i n g ( 1 - 3 . 1 1 ) t o repla c e the expression £ x$a9 we f i n a l l y have J = 1 n - 1 + 1 ] P x ( ^ a ) + < » ^ ) ( S s £ ) P x ( x o + e m ) ( 2 - 4 . 1 5 ) xm. ™x If we now repla c e x*a by x^+e^ and hence x m a by x t a a+^» w e o b t a i n n - 1 3 = 1 - [ ^ ( 1 + 1 / N X ) + 1 ] P x ( x a + e m ) + [ 1 / x + N X + X - + 1 . - ^ ~ ] . P ^ ( x a + 2 e m ) ( 2 - 4 . 1 6 ) If we know the r i t e r a t e d values of X, P l, P n_jL J t t i e 52 s u b s t i t u t i o n of (2-4.15) and (2-4.16) i n t o (2-4.12) and the r e s u l t i n t o (2-4.9) y i e l d s Then the ( r + l ) s t i t e r a t e d values of ..., p n may he found from (2-4.7) and (2-4.8). One suggestion f o r i n i t i a l estimates of \, p^, Pn_-j_ i s the moment e s t i m a t o r s . 2-5 Covariance M a t r i x of the Maximum L i k e l i h o o d Estimators A. Method of C a l c u l a t i o n As w i t h the Poisson-multinomial d i s t r i b u t i o n , d i r e c t c a l c u l a t i o n of -Q i s n e a r l y i m p o s s i b l e . We wish t o show however, th a t i t may be c a l c u l a t e d i n d i r e c t l y by the same method as i n § 1-5A, I.e. u s i n g Rao's theorem which i s s t a t e d i n t h a t s e c t i o n . To prove t h i s d i s t r i b u t i o n s a t i s f i e s Rao's theorem, the same procedure as i n § 1-5A i s f o l l o w e d . From the remarks at the beginning of 1-3 we conclude t h a t the f a c t o r i a l moment generating f u n c t i o n i s given by (2-2.6) where s i s set equal t o 1 i n s t e a d of 0. Since 1 - ̂  P i = P n > 0 , i t i s c l e a r t h a t ^ i = l g*("s) i s i n f i n i t e l y d i f f e r e n t i a b l e . Hence a l l the f a c t o r i a l moments are f i n i t e . Because each moment about the o r i g i n i s a f i n i t e l i n e a r combination of f a c t o r i a l moments, these moments are a l s o f i n i t e . Lemma 2-1. For the Poisson-negative m u l t i n o m i a l d i s t r i b u t i o n , n-1 n-1 1 1=1 1=1 53 Proof: Using the convention defined i n (2-1.3), P->(x+S ) = e'x V l ! ( X 1 + -' ' + X n - 1 + H z - 1 ) ! ( V " - - + X n - 1 + N z ) X m z l l z l ^'•.•.xn.1!(NZ-l)« x ^ l n " 1 x, p 1=1 n-1 ( fi . - i r« r (^i + '-'+X- T+HZ-1)! ) V x, P->(x) + We-X y ( x B / a I ) — — ( i = l 1 Z = l x 1 ! . . . x n _ 1 ! ( N Z - l ) : V 1 ! ' P n N Z ( T T P ^ 1 ) z ] (2-5.D 1=1 ) Let us observe t h a t f o r z _> 2 ( x ^ . . )J x 1 ! . . . X j ^ ! (Nz-1 )l (x x+.. .+x n_ 1+Nz-N-l)! _ J L (x x+.. .+x n_ 1+U Z-k) x 1 ! . . . x n _ 1 ! ( N Z - N - l ) l J J_ (Nz-k) n-1 < (^I+...+^I.I+NZ-N-I)I / r x ± + 1\N - x 1 ! . . . x n _ 1 J ( N Z - N - l ) ! { £ l We may use the above i n e q u a l i t y t o s u b s t i t u t e f o r the f a c t o r i a l s i n (2-5.1) f o r z=2, 3, ... . Thus P- f n ^ . (x., + . . .+X„ .,+N-l). P x ^ m ) < — 1 x i P 5 ( x ) + N\e - X - 1 S l l - V - M i t i ^ . . . . x ^ K N - l ) ! 54 i = l 1=1 z ^ ( z - l ) ! x 1 ! . . . X n . 1 ! ( W - l ) ! n-1 -) l ( " i , " n " » i ^ i i = i The second t e r n i n the "braces i s the "zasl" term i n the expansion of P-»(x). The t h i r d term i n the braces i s equal t o P^>(x) minus the "z«o" term i n i t s expansion, a f a c t which i s easily- seen i f we re p l a c e z by z-1. Since each term i n the expansion of P^(x) i s non-negative, we can conclude px<x+v) < i f i * i * * ( * > + ^ V 1 ( i = i n-1 + *n* ( 1 X i + 1 ) N N x PxWJ 1=1 I f we now note t h a t e~ x, p m , p n are p o s i t i v e and l e s s than one and x i J> 0 f o r a l l i , we can w r i t e n-1 n-1 v P x(3c+e m) <[ £ x i +HX [1 + ( Y X j + l f ] | k 1=1 i = l ' Q.E.D. Lemma 2-2. For the Poisson-negative m u l t i n o m i a l d i s t r i b u t i o n , E( |a/5X ( l o g L ) l 5 ) < ». 55 Proof; By the same argument as i n lemma 1-3, we are ahle t o o b t a i n (1-5-3) i . e . E(|£_ l o g L p ) < B^E (| -5 p ) ax P̂ OO S u b s t i t u t i n g from (2-4.3) f o r the d e r i v a t i v e A f t e r r e p l a c i n g the absolute value of the sum by the sum of the absolute values and u s i n g the r e s u l t of lemma 2-1, we have E ( l o g L p ) < B 3 E ( I Y V N U l +( Y x i + 1 ) N } f A X \ N X P M ( 1=1 1=1 J n-1 5 + (1/Nx) Y 3=1 Upon expansion t h i s w i l l be a f i n i t e sum of terms of the f o l l o w i n g type - n-1 constant . E ( x j ^) 3=1 i (2-5.2) where the n^ are non-negative Integers These terms are a l l f:Lnite since we know a l l the moments are f i n i t e . Hence the r e s u l t f o l l o w s . Q.E.D. 56 Lemma 2-3- For the Poisson-negative m u l t i n o m i a l d i s t r i b u t i o n , E(WhV± ( l o g L ) p ) < « f o r i = 1, 2, ..., n-1. Proof; By the same argument as i n lemma 1-3, hut r e p l a c i n g X by P i we get (1-5.3) w i t h p̂ ^ r e p l a c i n g x. Thus ' * -z % d/9p* [ P ^ ( x ) ] , E ( | l - l o g L p ) < B?E(] 1 x V P ) 3P ± P$(x) Le t us s u b s t i t u t e from (2-4.2) f o r the d e r i v a t i v e . rr nZ^ x.+l B-*(x+e. ) , < ^ , X i / P i + ( 1 / P n ) y ^ . ^ _ - | i - ^ p ) k=l p n p i p 3 ( x ) By the manipu l a t i o n of absolute value s i g n and use of lemma 2-1, we a r r i v e a t n-1 E( |A_ l o g L | 5 ) < B 5E ( x±/v± + ( l / p n ) £ ^ 9 p i k=l p n P ± ( 3=1 3=1 ; 1 I f t h i s expression i s expanded, a sum of terms l i k e those i n (2-5.2) i s obtained and by the same reasoning as t h e r e , the r e s u l t i s obtained. Q.E.D. 57 I f we consider Rao's theorem which i s s t a t e d near the end of §1-5A, we n o t i c e t h a t lemmas 2-2 and 2-3 show t h a t the Poisson- negative m u l t i n o m i a l d i s t r i b u t i o n s a t i s f i e s i t s c o n d i t i o n s by choosing h = 1. Hence (1-5-5) w i l l h o l d , and f o r samples of reasonable s i z e we may use the approximation where J * = I * I * XPT 1 * 1 * Pn X p ^ p n - l X P«_iPi ' n - l ^ l (2-5-3) I * ] X p n - 1 p l p n - l I * p n - l p n - l i s the i n f o r m a t i o n m a t r i x , i . e . (2-5.4) I s t * = E ( l o g L • ̂ ~ l o g L ) (2-5.5) as at where L i s the l i k e l i h o o d f u n c t i o n f o r the Poisson-negative m u l t i n o m i a l d i s t r i b u t i o n . B. C a l c u l a t i o n of the Elements of J * . Before proceeding w i t h the c a l c u l a t i o n , l e t us f i r s t prove the f o l l o w i n g lemma which w i l l be of use t o s i m p l i f y the n o t a t i o n . 58 Lemma 2-4. S * (x + l ) ( x + l ) P x ( x + e JP-Cx+e ) Define A. . = -1 + ) ... } — i J,,, » x — - i - xl=° ^ - 1 = ° p i p j Px-W (2-5.6) Then A ^ = A ^ = A, say i,j,k,m = 1, 2, ..., n-1 Proof; The p r o o f i s i d e n t i c a l t o t h a t of lemma 1-5 except t h a t i n p l a c e o f equation (1-4.7) we r e f e r t o (2-4.4). We must a l s o note t h a t P^ now r e f e r s t o the p r o b a b i l i t y f u n c t i o n of the Polsson-negative m u l t i n o m i a l whereas i n lemma 1-5 i t r e f e r r e d t o tha t of the Poisson-multinomial. Q.E.D. Consider I x x * . I f we s u b s t i t u t e (1-4.1) i n t o (2-5.5) and use the same argument as was used f o r o b t a i n i n g (1 -5.1$), we get a formula f o r I x x * which i s e x a c t l y the same as t h a t f o r 1^^ i n the l a s t mentioned equation. Now s u b s t i t u t e f o r the d e r i v a t i v e u s i n g (2-4.H). Then (1/P ) I . * - f ... f — ( P-(x+eV ) A« v L ^ P̂ (x) ] H\p x m 1 ̂ t l - 1 n-1 n - i i - [1 + (1/HX) I x±] P x(5c) i = l j Expansion of t h i s expression u s i n g the d e f i n i t i o n of ex p e c t a t i o n and lemma 2-4 y i e l d s n-1 2 ( 1 / P f c u * = A + 2 + (1/N 2X 2) E [ ( Y X±) ] 1=1 n-1 - (2/N 2X 2p m) E[( Y X ± - l ) X m ] - (2/NXpm) E f X j 1=1 n-1 + (2/NX) Y E ( x i ) L e t us make use of equations (2-3.; ) t o s u b s t i t u t e f o r the e x p e c t a t i o n s . A f t e r s i m p l i f i c a t i o n , we f i n d ( 1 / 6 ) I X X * - A - ( l - p n ) [ H ( X + l ) ( l + P n ) + 1] / HXP n 2 (2-5.7) S i m i l a r l y we may o b t a i n the other e n t r i e s of the i n f o r m a t i o n m a t r i x P j x \ NXA/p n + [(NX+N+l)p n^ - l / p n - 1X]/P n (l/p)I p * = N2X2A/pn2 + (NX/pn2)[-(Srx+N+l)/pn2 i + l/p n + NX + 1] f o r 1 4 J (1/P frplP * = K2X2A/pn2 + (NX/pn2 )[ - (HX+H+1 )/pn2 + 1/Pn + NX+ 1] + HX/pnPj_ (2-5.8) The c a l c u l a t i o n s of the above r e s u l t s are o u t l i n e i n Appendix 2B. Now l e t us d e f i n e 60 Bxx* = I * XX BxP* - (1/p) IXP 1 BPP* - (1/p) I p i p 3 4 i (2-5-9) Hence B * + Nx/p„p, pp n I = (1/P) I * (2-5.10) J * m B Bxx* BXP* B. * Xp Xp B *+N\/p P., B * PP XP B * PP B *+NX/p P , pp T A / p n F n - l Now, from (2-5.3) we have t h a t f\ - J - 1 , and u s i n g Appendix ID, formulas (1D-1) and (1D-6) w i t h the appropriate a s s o c i a t i o n o f v a r i a b l e s , n-1 det J* > B u * + (B u .B p p * - B,p.2)pn(l-Pn)/Hx} f t (-»-) i = l p n p i (2-5.11) 61 ! NX + B p p * P n ^ n ) var X sr — — ^ " P BXX # N X +( BXX* Bpp*- BXp*'' i ^ n t ^ P n ) B * p p„ * 0 BXX* N X +( BXX* BPP*- BXP* 2to B(i-p n) c o , , ^ , ^ , . - 1 ^ X x ' V - ^ X P ^ n V , S [ B u * H X + ( B x x * B p p * - B x p . 2 ) p n ( l - P n ) ] H X ~ l ( , ? n p i ^ X X ^ P P * - ^ * 2 ^ ! 2 var p. = £1 [:^--- ——=^ ^- ^—= 61 NX C B u * N X + ( B x x * B p p * - B X p * 2 ) p n ( l - . p n ) ] N X ( (2-5.12) C o r o l l a r i e s 2.1 and 2.2 i n Rao [19^7] s t a t e t h a t i f the d i s t r i b u t i o n s a t i s f i e s lemmas 2-2 and 2-3, then the maximum l i k e l i - hood estimators are minimum variance estimators f o r l a r g e samples and i n terms of the g e n e r a l i z e d v a r i a n c e , det f l * , are asym- t o t i c a l l y e f f i c i e n t . 2-6 E f f i c i e n c y of Method of Moments f A. Method of C a l c u l a t i o n The method used i s i d e n t i c a l t o the one described i n 1-6A. To d i s t i n g u i s h c e r t a i n q u a n t i t i e s such as the Information m a t r i x , covarianee m a t r i x , e t c . f o r the d i s t r i b u t i o n now under consider- a t i o n from those f o r the dis-tribution described i n chapter 1, s u p e r s c r i p t s t a r s w i l l toe w r i t t e n a f t e r the symbols (e.g. fi *, J * , M*, e t c . ) . Thus the e f f i c i e n c y i s given by 62 E f f = (2-6.1) det n * det J * where PL * I s the covariance m a t r i x of the moment es t i m a t o r s . As I s shown i n Appendix I E , det A * = (det ( j * ) 2 det M* (2-6.2) where det (f* i s given by (1-6.6) and M* i s given by (1-6.4) except t h a t the q u a n t i t i e s now r e f e r t o the Poisson-negative m u l t i n o m i a l d i s t r i b u t i o n . B. C a l c u l a t i o n of det M* i n Terms of the Parameters We may repeat the argument t h a t l e d t o (1-6.7) f o r the present d i s t r i b u t i o n and^obtain an i d e n t i c a l r e s u l t , namely var X± = ( l / p ) [ E ( X ± 2 ) - E 2 ( X ± ) ] S u b s t i t u t i n g f o r the expectations u s i n g (2-3.3), we o b t a i n var X ± - ( l / P ) ( N X p i / p n ) [ ( p i / p n ) ( N + l ) + l ] (2-6.3) I f we repeat the argument t h a t l e d t o (1-6.9) we w i l l o b t a i n cov ( X ^ X j ) = ( l / p ) [ E ( X ± X j ) - E ( X ± ) E(X^)3 and s u b s t i t u t i o n from (2-3-3) y i e l d s c o v p C ^ ) = (l/p)N(N+l)X P i P j / P n 2 (2-6.4) S i m i l a r l y , r e p e t i t i o n of the arguments l e a d i n g t o (1-6.11) and (1-6.15) l e a d t o i d e n t i c a l equations f o r the Poisson-negative m u l t i n o m i a l d i s t r i b u t i o n . By s u b s t i t u t i n g (2-3.3) i n t o (1-6.11) 63 and (1-6.15) and u s i n g Appendices I F and 1G w i t h G ±* r e p l a c i n g Gj^ and P k / P n r e p l a c i n g p f c, we have r e s p e c t i v e l y cov (W 2,Y k) = (1/0)(N\ P l / p n ) H X * 1-P„ f 1-Pr, ") where H * = 2 I G 2* ( 2) + 3G 1* • p n I p n i 1-p^ _ NX[G * ( £) + i ] + 1 p n (2-6.5) and var = ( l / P ) [ N X ( l - p n ) / p n ] H 2 * l - p _ 3 l-p„ 2 1-p where H 2* = G^* ( 2) + 6G 2* ( 2) + 7 G l * ( S) r n * n 1-P„ l - p _ 2 + l-NX ( 2) [ G ] L* ( E) + i ] n n n (2-6.6) Now l e t us s t u h s t l t u t e (2-6.3) through (2-6.6) i n t o the expression f o r M* which i s given by (1-6.4). Then N X ( l - P n ) H * N2 NXP, n NXpn NXp. n-1 'n -RJ* n NXpn p, ±[_J: (N+l)+l] p n p n P l P2 H(N+1 )X-^-7y p. n p l p n - l N(N+1 )x-^-£ n NXp n-1 H * n PT P i N(N+1 )X 1 g n NXP W T P„ T _J^ll[^zi(H+l)+l] p n p n y (2-6.7) 64 The determinant of the above m a t r i x i s found i n Appendix 1H i f we re p l a c e H ±, N», and R by H ±*, (N+l)/p n, and NX/p n r e s p e c t i v e l y i n the r e s u l t g iven i n the appendix. Thus det M = (NX/BpJ ( l ^ K f l P t J l H ^ C l + C l ^ K N + l J / p J i = l \ * 2 \ (2-6.8) C. Determination of the Jacoblan, det 4* The expression f o r (}* i s the same as the one given f o r (J i n (1-6.21) w i t h the exception t h a t the q u a n t i t i e s r e f e r now to the Poisson-negative m u l t i n o m i a l d i s t r i b u t i o n . The p a r t i a l d e r i v a t i v e s i n (1-6.21) can be obtained i n a s t r a i g h t f o r w a r d manner from (2-3.9) and (2-3.10). Hence, by d i f f e r e n t i a t i n g , ,2, ax _ N+l E c^¥) 9E(¥5) H [E(W2)-E2(¥)-E(¥)]2 aX m N+l . E(W)[2E(W2)-E(W) j aE(X±) N [E(W2)-E2(¥)-E(W)]2 ap± (N+l)E(X±) aE(¥2) [E(W 2 ) -E 2(¥)+NE(¥)] 2 ap* E(X . ) r o o <? 1 2 ^-^T 2 -E(¥^)[E(¥^)-E^(¥)+NE(¥)] 3E(Xj) E^(¥)[E(¥^)-E^(¥)+NE(¥)] + E(¥)[E(¥2)-E(¥)][E(¥)-N] - E5(W)| i f i=fj 3P4 E(X. ) ( o o o 1 - ^ - i - ^ 2 ) -E(¥^)[E(¥^)-E^(¥)+NE(¥)] aE(X±) E';(¥)[E(¥'::)-E';(¥)+NE(¥)]i 65 + E ( W ) [ E ( W 2 ) - E ( W ) ] [ E ( W ) - N] - + E ( W 2 ) - E 2 ( W ) - E ( W ) E ( W ) [ E ( W 2 ) - E 2 ( W ) + N E ( W ) ] E 5 ( W ) | Let us use (2-3.3), (2-3.6), and (2-3.7) t o s u b s t i t u t e f o r the expectations and then s i m p l i f y the r e s u l t s t o o b t a i n p n S E ( W ^ ) N(N+l)(l-p n) 2 ax FPn 9E ( X I ) N(N+l)(l-p n) 2 P i p n 3 aE(w 2) N(H+l)x(l-p n) 2 ap± Dp ±P n aE(xd) N(N+l)x(l-P n) 2 aPi ^ i P n B E ( X 1 ) N(N+!)X(1-Pn)2 f o r i4j IB. NX (2-6.9) where P and D are de f i n e d by P = 2(l-p n)[N(x+l)+l] + p n D = - N(l-p 2 ) - 1 - - i - [ 2 N 2 X p n ( l - P n ) n N+l n n - N 2 X 2 ( l - P n ) 2 + N X ( l - P n 2 ) ] (2-6.10) We may now s u b s t i t u t e these values i n t o (1-6.21) and o b t a i n an e x p l i c i t e xpression f o r G;*. The determinant of (f* i s c a l c - 66 u l a t e d i n Appendix 2C i f we replace Q i n the appendix by N ( N + l ) ( l - p n ) 2 / p n . The r e s u l t i s , + ri* P n n + 1 t ( N + 1 ) ( 1 - P n ) + D + F P n ] det Q> = - -2 _ £ Cp-6 11) Wow we are able t o s u b s t i t u t e (2-6.8) and (2-6.11) i n t o (2-6.2) t o get aet A. . C + 2^(« +i)(i-P n)^ a1 2 r _ 8 ( N + i r N ( N X ) n - - L ( l - p n ) V 1 2 1 n-1 (2-6.12) + ( l - P n ) H 2 * ( N + l ) / p n | 7 j " P i ) 1=1 By the same argument t h a t l e d t o (1-6.2), i t i s easy t o see tha t 1 E f f = d e t i \ * det J * I f we repla c e the determinants by t h e i r e x p l i c i t -expressions given i n (2-5.11) and (2-6.12), we f i n d t h a t N 2 ( U + l ) 4 ( l - p )5 E f f = — ^ — — p n ^ X X ^ P n ^ P n X BXX* Bpp*- BXp* >] (2-6.13) A . 1 [ (N+l ) ( l - p n )+D+Fp n]* [ (H^-B^*'"' ) p n + ( l - P n )(N+1 )H 2* ] 2-7 Sample Zero Frequency and U n i t Sample Frequency Estimators A. Sample Zero Frequency and F i r s t Moments This type of e s t i m a t i o n i s u s e f u l under the same c o n d i t i o n s as o u t l i n e d i n ^ 1-7A, i . e . the zero sample occurs f a i r l y f r e q u - 67 e n t l y . S e t t i n g X = 0 i n (2-2.10), we o b t a i n P^(0) = exp [ X ( p n N - l ) ] (2-7.1) I f we d e f i n e P ( a ) i n the same manner as i n ^ 1-7A, then E [ | F ( 3 ) ] = P-("C) = e x p [ x ( p n N - l ) ] and hence ( l / p ) P ( 0 ) i s an unbiased estimator f o r P_*(6). Thus, t o o b t a i n the sample zero frequency estimators X and p^ of X and p^, we solve the equation (1/B )F(0) = exp [T(p ^-1)3 along w i t h the equations of (2-3.3) which i n v o l v e the f i r s t moments. Hence we solve the set (1 / P)P(0) = e x p t ^ p Z - l ) ] T± = EX V ± / v n i = 1, 2, n-1 (2-7.2) f o r X and p^ . To do t h i s we add the second equation of (2-7.2) f o r 1 = 1, 2, , n-1. Then we have i = l NX and hence p n = n-1 (2-7.3) + NX i = l We may now s u b s t i t u t e t h i s q u a n t i t y i n t o the l o g a r i t h m of the f i r s t equation of (2-7.2) and f i n d N B / N X \ n-1 y x,+NX \ i = l 1 I - 1 68 -We must now solve f o r x\ I t i s best t o use some numerical procedure. Once having done t h i s we may f i n d p n from (2-7-3) and f i n a l l y p^ from (2-7.2). B. U n i t Sample Frequency E s t i m a t i o n For cases where the u n i t samples occur f r e q u e n t l y , the u n i t sample estimators are sometimes u s e f u l . From (2-2.7) P x ( r k ) = i i K H BS k = exp [ x ( p n N - l ) ] N p n N p k X (2-7.4). s=o Now E[ (1/6 )F(e* k)] = P - ( e k ) and thus the u n i t frequency estimator i s unbiased f o r p ^ ( e k ) . To f i n d the u n i t sample estimators x and p^ f o r x and p^ r e s p e c t i v e l y , we must solve (l/p)F(eT k) = e x p [ x ( ? n H - l ) ] ^ n \ X (2-7.5) together w i t h the zero sample estimator (1/6)F(0) = exp [ x ( p n K - l ) ] (2-7.6) A f t e r d i v i d i n g (2-7-5) by (2-7-5) w i t h k = l , we get P(®v) P k (2-7-7) F ( % ) P X A l s o , a f t e r d i v i d i n g (2-7-5) w i t h k=l by the f i r s t equation of (2-7.2) and n o t i n g the d e f i n i t i o n of p n n-1 N F ^ J / F ^ ) = H ( l - £ P k ) P i * (2-7.8) k=l 69 I f we now take the l o g a r i t h m of (2-7.6) and s u b s t i t u t e f o r the p k from (2-7.7) ^ n-1 l l o g l & l + 1 = [1 - - 4 - Y F(&_)] (2-7.9) X P p < e l > M L L e t us d i v i d e (2-7.8) by X and s u b s t i t u t e f o r p k , k = 2, 3, . . n - 1 from (2-7-7) 1 / X = J ~ _ [ 1 - — L _ V P ( e k ) ] (2-7.10) F ( e x ) F ( e - ) k ^ * We may now s u b s t i t u t e t h i s i n t o (2-7.9) and o b t a i n p,NF(0) - p n nZ^ ^ N 1 v [1 - — k - V F(e, )] l o g l i ^ i + l F ( e l > F ( e l > k=l k 6 P i V " - N = Cl - — 7 F(e )] F ( e l ) k = l k T h i s may be r e w r i t t e n as P i ' ^ N PiNF(O) t?(n) [1 - — ± — 7 F ( e k ) ] [ 1 V l o g li21 - 1] + 1 = 0 (2-7.11) Probably the best way t o solve t h i s f o r p^ i s t o use v a s u i t a b l e numerical procedure. A f t e r doing t h i s , X may be obtained from (2-7.10) and p k , k=2, 3, n-1 from (2-7-7). 70 CHAPTER I I I LIMITING DISTRIBUTIONS OF THE POISSOSi-MULTINOMIAL AND POISSON-NEGATIVE MULTINOMIAL DISTRIBUTIONS 3-1 I n t r o d u c t i o n I n many a p p l i c a t i o n s c e r t a i n parameters may "be known already t o "be very l a r g e , t o be a p a r t i c u l a r value, or t o be almost n e g l i g a b l e . U s u a l l y , I f circumstances permit, i t i s much e a s i e r t o consider the l i m i t i n g d i s t r i b u t i o n s as the parameters approach t h e i r r e s p e c t i v e l i m i t s . I f a p a r t i c u l a r p^ i s allowed t o approach zero i n e i t h e r of the above d i s t r i b u t i o n s , %fee- form e£ %fee d i o t r i b u t l o n c , the form of the d i s t r i b u t i o n w i l l remain unchanged except t h a t the I* 1 v a r i a t e w i l l be completely ignored. 3-2 The Poisson-Polsson D i s t r i b u t i o n The most i n t e r e s t i n g l i m i t i n g d i s t r i b u t i o n i s the one i n which N -» « and p^-» 0 f o r i = l , 2, n-1 ±n such a way th a t Np^=a^ = constant. Then, by u s i n g the f a c t t h a t n-1 we have from (1-2.1) 1=1 n-1 N ' a±( s i - l )+l] -X 1=1 I f we now apply the mathematical i d e n t i t y l i m (1+Q/N) = l i m (1+l/ST) (3-2.1) N* oo N->oo 71 we w i l l have g L ( s ) = exp 1 X exp [ £ a 1 ( s 1 - l ) ] - x| 1=1 I f , i n s t e a d , we s t a r t from (2-2.6), we get P g * ( s ) = l i r a g*(s) = exp ( X l i m (3-2.2) n nT[ 1 " I s i ? i 1=1 By u s i n g the d e f i n i t i o n of the and a f a m i l i a r theorem about l i m i t s , = exp f X l i m [1-(1/N) V a ± ] l i m [1-(1/N) ^ a ^ ] - Xf F i n a l l y , l e t us "apply i d e n t i t y (3-2.1) to tne above l i m i t s . A f t e r s l i g h t s i m p l i f i c a t i o n we get n-1 S L * ( s ) = exp i x exp[ £ a i ( s i - l ) ] - x| 1=1 (3-2.3) Equations (3-2.2) and (3-2.3) show t h a t S L ( s ) = g L * ( s ) . Hence both the Poissson-multinomial and Poisson-negative multinomial have the same l i m i t i n g d i s t r i b u t i o n . From the form of the p r o b a b i l i t y generating f u n c t i o n we see tha t t h i s i s a Poisson- m u l t i v a r i a t e P o i s s o n , or the m u l t i v a r i a t e analogue of the Neyman Type A d i s t r i b u t i o n . From what has been done i n cnapters I and I I i t i s , f o r the most p a r t , an easy matter to o b t a i n the same r e s u l t s f o r the l i m i t i n g d i s t r i b u t i o n as we obtained f o r the two previous d i s t r i - b u t i o n s . We simply a l l o w the parameters to approach t h e i r l i m i t s i n the formulas which give the q u a n t i t i e s we wisn to f i n d . The moments of the d i s t r i b u t i o n may be found from (1-3.3), 72 E(Xj.) = l i m NX p ± = X a ± , or from (2-3.3), N-* E ( X ± ) = l i m N X p i / p n = X a± S i m i l a r l y , E(X±2) = x ( X + l ) o ± 2 + Xa ± ^ (3-2.4) E ( X ± X j ) = X(X+l)a j.a^ Here we may consider e s t i m a t i n g the parameters. We must he aware t h a t these are now X, a,, — , CL, n . From e i t h e r x n—x (1-3.9) and (1-3.10), or (2-3-9) and (2-3-10), we can de r i v e the moment estimators X = ^ E(W 2) .2/ E(W*)-E*(W)-E(W) o o t (3-2.5) E(X. )[E(W*)-E*(W)-E(W)] ' a. = l i m Np. - — — — — — 1 N̂ co 1 E^W) A l s o , the maximum l i k e l i h o o d e stimators f o r the may he found from (1-4.11) or (2-4.8) by t a k i n g the l i m i t of Hp, as N^oo. Thus we have The maximum l i k e l i h o o d estimator f o r X i s given by e i t h e r (1-4.14) or (2-4.11) since both of these remain unchanged as N-^oo. We must r e a l i z e , however, thax now r e f e r s t o the -ft d e n s i t y of the l i m i t i n g d i s t r i b u t i o n . We s h a l l henceforth denote the l i m i t i n g d e n s i t y by . 3-3 The Information M a t r i x Prom e i t h e r (1-5.12) or (2-5.6) I t i s c l e a r that i n the l i m i t i n g case • * (x±+l)(x +l)lx(x+e±)Zt(x+e ) A = -1 + ) . . . ) * • »— L e t J denote the i n f o r m a t i o n m a t r i x and I ̂  denote i t s e n t r i e s . Then I . . = l i m I = l i m I * Prom (1-5.16) and (1-5.18), or (2-5-7) and (2-5-9) (1/P) I = l i m B.. = l i m B * = A (3-3-1) '~ A A _ A. A. TVT v „ A A I n c a l c u l a t i n g the other e n t r i e s we must he c a r e f u l since corresponds t o Np̂ ^ r a t h e r than p^. Thus from the d e f i n i - t i o n of the i n f o r m a t i o n m a t r i x , (1-5.2) xal Brf (x) aX 9 a i x mmmmJ\. = l i m E [ — , i r - P-(x) — 2 P^(x)3 N * * P-"(X) ax a(N P j L) x = l i m (1/N) E[ % 1- P-(*)£- P - ( x ) ] N * « P - d{x) ax = l i m ( I _ /N) = l i m ( I . _ */N) Now from (1-5.17) and (I-5.18), or (2-5-8) and (2-5-9) ( 1 / 3 ) 1 = l i m (B /I) = l i m (B */N) = -XA+1 ~ x a i &+• A p W-̂oo x p (3-3 .2) S i m i l a r l y , u s i n g the same equations, ( 1 / p £ a a " < 1 / F 3 ) L L M D A 2 ) = ( V 3 ) l i m (1L » / N 2 ) ^ i 3 i>« p i p j N->» p i p j = l i m (B/N 2) = l i m ( B * / N 2 ) PP PP = X(\A - 1 ) ( 1 / 3 = X ( X A - l + l / a i ) A l s o , Hence we may summarize J as J = -XX -Xp. I I i x p 1 % l P l I x p n - 1 ~ p l p n - l I XP, n-1 " p l p n - l ~ p n - l p n - l ( 3 - 3 . 3 ) ( 3 - 3 . 4 ) where ( 1 / 3 ) 1 2xx - A = B. XP, -XP ( 1 / 3 ) 1 ^ . , = 1 l p i p 3 •PP - XA+1 X(XA-l) i f 143 ( l / p ) I p = B p p + \/a± = x(xA-l+l/ a i) (3-3-5) 75 3-4 E f f i c i e n c y of Method of Moments By u s i n g the same arguments as are used t o o b t a i n (1-6.2) and (1-6.5)> we can show t h a t E f f « J: (3-4.1) (det (j) det M det J where M i s the covariance m a t r i x of the moment estimators of \, a^, c&n_x and det (f i s the Jacobian 9[E(¥if),E(X1),...,E(Xn_1)3 F i r s t l e t us consider M. M may be found by t a k i n g l i m i t s i n e i t h e r (1-6.19) or (2-6.7). This r e s u l t s i n cov O ^ X j ) = (1/3) \a±a.j f o r ±4j \ var X± = (1/3) Xa i(a i+1) _ ' (3-4.2) cov (¥ 2,X i) = (1/3) X a ^ var W2 = (1/3) Xa H 2 where n-1 a = Z aJ "1 i = l 76 Si l i m EL = N->« 1 l i m H 1* = (2X+l)a + (4x+3)a + 1 l i m H 2 = l i m H 2* (3-4.3) N-»o N->< a3(4x2+6x+l )+2a2(2x2+3x+3)+a(6X+7)+l Because each entry of M and M* converges t o the corresponding entry of M as N->», det M = l i m det = l i m det M* (3-4.4) Now consider det 9 [ X $ ct-i $ • • • 3 Q5 •! 3 det <| = l i m x n"-L N->» a[E(W^ ),E(X X ) s . . . ,2(3^^)3 = l i m N->» _a_x 3E(W ) q " 9E(¥c:) aa 'n-1 ax aE(xx) aa 'n-1 aE(tr) aE(xx) acu aE(x1) aE(x2) ax ao^ aa. n-1 9 E ( X n - l ) By v i r t u e of (1-6.21) and the f a c t t h a t a i=Np i, 77 det (£ = l i m N 1 1 det (| = l i m i f 1 " 1 det ^ (3-4.5) From ^ 3-3 we see t h a t det J = l i m N»» XX I XP A n-1 "P 1P 1 A ' P l p n - 1 i.„ A 7 i„ „ A ' X pn - 1 P n - l p l ^ n - l P n - i l i m H " 2 ( n _ 1 ) det J (3-4.6) The same r e l a t i o n holds i f we consider J * . Now i f we s u b s t i t u t e (3-4.4), (3-4.5) and (3-4.6) i n t o (3-4.1) we have E f f m 5 ( l i m N n _ 1 d e t (|) 2(lim det M) I N->» »••?>» . ( l i m j j - 2 ( n - l ) d e t J ) ) N*« ) = l i m j (det (J) det M det J N̂ co ( = l i m ( E f f ) N-*» -1 (3-4.7) E x a c t l y the same r e s u l t w i l l h o ld i f we consider (}*, M*, and J * i n s t e a d of (f, M, and J. By t a k i n g l i m i t s e i t h e r i n (1-6.24) or In (2-6.13) and then s u b s t i t u t i n g f o r the B's from (3-3.5), 78 E f f = 2 — (3-4.8) [XA(a+l)-a][H 2(a+l J-H^ ] where and Hg are given by (3-4.3). This c a l c u l a t i o n i s o u t l i n e d i n Appendix 3A. 3-5 Sample Zero Frequency and F i r s t Moment Estimators From e i t h e r (1-7.1) or (2-7.1) we can deduce t h a t the p r o b a b i l i t y of the zero sample i s P-(tf) = l i m exp [ X ( p r N - l ) 3 = exp [ X ( l i m (p N ) -1)]. Because a = N ( l - p n ) , we have l i m p / = l i m ( l - a / N ) N = e" a (3-5-1) Thus P-(0) = exp [X(e" a-1)3 D e f i n i n g F ( a ) t o be the frequency w i t h which "t = a = ( a ^ , ..., a n - i ) occurs i n B obs e r v a t i o n s , we have E(|F(0)) = P x(0) = exp [X(e" a-1)3 (3-5-2) Thus (l/p)F(0) i s an unbiased estimator f o r P (0). The sample zero frequency estimators X and p^ f o r X and p^ r e s p e c t i v e l y may be found by s o l v i n g the above equations simultaneously w i t h the f i r s t moment equations of (3-2.4), i . e . 79 (1/6)F(0) = e x p [ t ( e - a ' - l ) ] (3-5.3) Adding the second equation of (3-5-3) f o r 1=1, 2, ..., n-1, n-1 n-1 1=1 i = l Thi s equation may he solved f o r a and used t o replace a i n the f i r s t equation of (3-5-3)- Hence (1/B)F(0) = exp X e x p [ ( - l / X ) £ T± - X \ 1 ^ i = l This may be solved f o r X u s i n g a s u i t a b l e numerical method. Once having done t h i s we can solve f o r the 3̂  u s i n g the second equation of (3-5-2), i . e . - X^/T. 3-6 U n i t Sample Frequency E s t i m a t i o n 'Taking l i m i t s i n e i t h e r (1-7-7) or (2-7.4) w i t h the help of (3-5.1) y i e l d s P - ( e k ) = e x p [ x ( e ~ a - l ) ] a^Xe'** k = l , 2, ..., n-1 (3-6.1) Le t us note t h a t E [ ( l / 8 ) F ( x ) j = P - ( x ) , hence ( l / B ) F ( e k ) i s an unbiased estimator f o r £ x ( e ^ ) - Thus, i f we l e t X and be the u n i t sample estimators f o r X and r e s p e c t i v e l y , then we may solve (3-6.1) w i t h e k ) replaced, by i t s estimator, and (3-5.2) w i t h P^(0) replaced by i t s estimator f o r X and 80 c^, k = l , 2, ..., n-1, i . e . we must solve the system (1/P)P(0) = exp [x(e-«-l)] ( l / B ) P ( t k ) = exp Me-"-!)]^* ( (3-6.2) k = l , 2,..., n-1 Upon d i v i s i o n of the second equation hy the second equation w i t h k = l , we o b t a i n ^ 2 - °£ (5-6.,) and upon d i v i s i o n of the second equation by the f i r s t and s e t t i n g k = l , Ffe., ) , * = ct-i Xe" a (3-6.4) F(3) 1 By t a k i n g the l o g a r i t h m of the f i r s t equation of (3-6.2) and s u b s t i t u t i n g f o r the i n a = c ^ f ... + from (3-6.3) we o b t a i n (1/X) l o g - exp \ - - L - £ P ( e k ) | -1 (3-6.5) 6 1 F ( e l k ) ¥e may solve (3-6.4) f o r 1/X and then replace the ô . i n a usi n g (3-6.3) t o o b t a i n v a,F(0) (- a, „ , 1/X = -± exp ) A- V P(e. 81 Let us now s u b s t i t u t e t h i s equation f o r 1/x i n (3-6.5). Then \hz& i o g H O I . 1 ) « p (. A . T F ( e k ) \ + i = o 1 p ( * i > s i 1 F ( g i ) k = i k j Because t h i s equation does not lend i t s e l f e a s i l y t o exact s o l u - t i o n , i t i s best t o t r y a numerical procedure t o f i n d a^. Then ctp, a , may be found from (3-6.3) and X from (3-6.4). 82 APPENDIX 1A OBTAINING AN EXPLICIT EXPRESSION FOR THE PROBABILITY FUNCTION We w i l l s t a r t w i t h the p r o b a b i l i t y generating f u n c t i o n given by (1-2.1). g ( t ) = e X ( T N - l ) n-1 where T = ^ s i p ± + p n 1=1 Then D k g ( s ) = Xg(s) D j j T ^ ) (1A-1) For s i m p l i c i t y l e t Xg(s) = A, D k(T^) = B. Then ( 1A-1) becomes \&(b) = AB*. To f i n d the higher order d e r i v a t i v e s , l e t us r e s o r t t o L e i b n i t z Rule which s t a t e s t h a t x n - l x n - l , f l T 3v r / X n - l x ^ y n-l„ ^ ^ - l ' ^ n - l , Dn-1 <*» - I ( y 1 1 ^ V l " " ^ ' D n - 1 yn-i-° n" Now, we may d i f f e r e n t i a t e termwlse w i t h respect t o s 2 9 1 1 ( 1 o b t a i n Use L e i b n i t z Rule on each term. yn-l =° yn-2=0 . r n n-2 y n - 2 n x n - l y n - l p l 83 The sums are independent of each other and consequently we may reverse t h e i r order 3?n""2 "̂ n 1 r* f7 A ^ W ^ - I N „ yn-2T 4 y n - l y n-2=° y n - i = o n-2 'n-1 n " 2 V l D  x n-2" y n-2 D x n - l " y n - l E n-2 n-1 Continuing t h i s process we f i n d t h a t , a f t e r r e p l a c i n g A and B by the expressions they represent, x l x n - l yl=° yn-l=° . n y n - l n y l r , f f / - U i i x n - l " y n - l „ x l " y l n / J ^ n-1 [ x g ( s ) j D n - 1 ...D1 D k(T ; (1A-2) The remaining problem i s t o f i n d the above m u l t i p l e d e r i v a t i v e of T N. Not i c e D k ( T K ) = NT 5 1" 1 DfcT = NT 1" 1 p k Continuing, D x X l " y i ( \ T H ) = N p k D 1 X l " y i ( T H - 1 ) N-(x, -y, ) - l x..-y, = p k N ( N - l ) ( l - 2 ) ... [ N - ( x 1 - y 1 ) ] T X p i We may proceed i n the same manner through the d e r i v a t i v e s w i t h respect t o a l l the s^. The r e s u l t i n g expression i s 84 D ^ - l ^ n - l n x l - y l n-1 x _ y n-1 ^ pk ( T T P i 1 1 ) ^ ( N - 1 ) ... [N- £ ( x i - y i ) ] T i i i = l i = l n-1 n _ r ^ x i " y i ) _ i (1A-3) From the d e f i n i t i o n of a p r o b a b i l i t y generating f u n c t i o n , P x ^ k ) 5=1 ( V i ) T T V i = l s=0 a n d , s u b s t i t u t i n g from (lA.j-2), * n - l x n - l n-1 (x f e+l) I T *A • yx-o yn-1-o i = i i = i n-1 x n - l " y n - l _ x l " y l I f we use (1A-3) t o evaluate the second square bracket X "n=T c l * n - l n-1 „ n-1 r IT (ŷ )t"fT y±! ̂(y)3 ( V 1 ) T T x i ! ^1=° y n - i = ° 1 = 1 1 1 = 1 i = l n-1 x . - y . ^ N- V ( x . - y . ) - l i = i 1=1 F i n a l l y , we can express the combinations as f a c t o r i a l s and, a f t e r rearranging the terms, we a r r i v e at 85 V y,=o y« .=o i = l N-1 x l x n - l n-1 yl=° yn-l=° n-1 1=1 p n (1A-4) ( x i - y i ) 1 I f , as i n the case of the Poisson-multinomial d i s t r i - b u t i o n , N i s a p o s i t i v e i n t e g e r , then P^(x+e k) = /  X p k p n r - i x i B - l Nl u /-> n^T V 1 ^1=° y n - i = ° [N- £ ( x i - y i ) - i ] i i = l n " 1 p 4 X i " y i n-1 [ T T ) " 1 . T J Pg(y) i f N> £ ( x ± - y i ) 1=1 p n (X±-Y±)' 1=1 0 otherwise 86 APPENDIX IB CALCULATION OP MOMENTS PROM FACTORIAL CUMULANT GENERATING FUNCTION From (1-3.1), the cumulant generating f u n c t i o n f o r the Poisson-multinomial d i s t r i b u t i o n i s n " 1 N c ( s ) = x [ [ I s i P i + p n ] - l ] i = l For s i m p l i c i t y of n o t a t i o n , i n the f o l l o w i n g c a l c u l a t i o n s the expectations of products of X 1, X g, X^, X^ w i l l be c a l c u l a t e d e x p l i c i t e l y . I t i s c l e a r , however t h a t these r e s u l t s may be g e n e r a l i z e d t o products of any X^'s. Let us now c a l c u l a t e a l l the p a r t i a l d e r i v a t i v e s of c(s') w i t h respect t o the s.̂  of order l e s s than or equal t o f o u r . These are n ~ 1 N-1 1' TT = ^ I s i p i + p n > p l B s l i = l 2 n-1 N-2 2L£ 2 - XN(N-I)( I s i P l + P n ) P l 2 s s l 1=1 9 S 1 3 i = l XN(N-l)(N-2)( £ s l P l + p n ) P l 5 4 n _ 1 N-4 h 4. = XN(N-l)(N-2)(N-3)( £ s i P i + p n ) P l 4 * s l i = l 2 n ~ l u_2 5- Trh" X M ( H - 1 ) ( 2, s i p i + p n ) P i p 2 d s 2 S s l 1=1 87 6. d g = XN(H-l)(N-2)( ^ s i P i + P n ) P i P 2 a V s l 1=1 7. 5 ° 3 = XN(N-l)(N - 2)(N-3)( £ s l P i + p n ) p x 5 p 2 B s 2 3 s l 1=1 -3 P n p 1 N-3 8. L-S \N(N-l)(N-2)( ^ s i P l + p n ) p l P 2 p 3 BS^QSgdS^ 1—1 4 n _ 1 N-4 9 . L-2 2= XN(N-l)(N - 2)(N-3)( £ s i P i + P n ) P i 2 P 2 P 3 a s 3 9 s 2 9 s l 1=1 4 1 1 - 1 N-4 10. = XN(N-l)(N-2)(N-3)( £ s l P i + p n ) P X 2 P 2 2 4 n _ 1 u_4 11. L-£ = XH(N-l)(N-2)(N-3)( £ s i P i + p n ) P 1 P 2 P 3 P 4 as^as^as 2 as x i = i To o b t a i n the f a c t o r i a l cumulants corresponding t o the above d e r i v a t i v e s we set s=l. We denote the f a c t o r i a l cumulants by K i j k m w n e r e t h i s symbol represents the i t h cumulant w i t h respect t o X.^ -3th w i t h respect X 2, k t h w i t h respect t o Xy m t h w i t h respect t o X^. I f the l a s t s u b s c r i p t s are zero, they may be omitted (e.g. ~ ̂ 110 ~ E n )• Thus formulas 1-11 become r e s p e c t i v e l y 1. K x = NXp̂ ^ 2. K 2 = N(N-l) Xp ]_ 2 3. K 3 = N(N-l)(N - 2 ) X p x 3 88 4. = N(N-l)(N-2)(N-3) X P 1 4 5. K 1 ± = H(H-l) \ p x p 2 6. K 2 1 « N(N-l)(N-2) X P A 2 p 2 7. K 3 ± = N(N-l)(N-2)(N-3) X P A 5 P 2 8. K l x l = N(N-l)(N-2) X P l P 2 P 3 9- K 2 1 ]_= N(N-l)(N-2)(N-3) X P j S g P ^ 10. K 2 2 = N(N-l)(N-2)(N-3) X p x 2 p 2 2 11. N(N-l)(N-2)(H-3) X p - j P ^ p ^ ( r ) ( r ) Now de f i n e E ^ 1 ... X t * ) t o be the r ^ * 1 f a c t o r i a l moment w i t h respect t o X^, , and the r ^ f a c t o r i a l moment w i t h respect t o X^. Then, u s i n g t a b l e s converting f a c t o r i a l cumulants t o f a c t o r i a l moments, we have i f we def i n e the G^ by (1-3.2) 1. E ( X X ) = K x = NXP X 2. E(X^2h m K2+K±2 - N X p 1 2 [ N ( X + l ) - l ] = NXP 1 2G 1 3. E ( X X X 2 ) = K i ; L + K 1 0 K 0 1 = NX p 1 P 2 [ N ( X + l ) - l ] - N X p ^ p ^ 4. E ( X X ( 5 ^ ) = + 3K 2K 1 + K x 5 m N X p x 5 [ N 2 ( X2+3X+1 )-3N( X+l )+2 ] = N X P l 3 G 2 89' 5. E ( X l ( 2 ) x 2 ) = K ^ ^ + S K ^ K ^ o ^ N X p 1 2 p 2 G 2 6. E ( X 1 X 2 X 3 ) « K l l l + K H O K 0 0 1 + K 1 0 1 K 0 1 0 + K 0 1 1 K 1 0 0 + K 1 0 0K 0 1 0K 0 0 1 4 = NXpjP^PjOg 7. E ^ 4 ) ) = + 4-K^ + 3 K 2 2 + o K ^ 2 + K x a NXp x 4[N 5(X 5+6x 2+7X+l)-6u 2(X 2+?X+1) + llN(x+l)-6] 8. E ( X l ( 5 ) X 2 ) = K 3 1 + 3 K 2 1 K 1 Q + K 5 0 K 0 1 + 3 K n K 2 0 = N X p 1 5 P 2 G 3 9. E ( X ( 2 ) x 2 2 ) ) = K 2 2 + 2 K 1 2 K 1 0 + 2K21KQ1 + ZK^2 + + S A o 2 + 4 K 1 1 K 1 0 K 0 1 + K 2 0 K 0 1 2 + ^icfoi = N X P 1 2 P 2 2 G ? 10. E f X ^ ^ X j ) = K 2 1 1 + SK^i^oo + K210 K001 + K201 K010 2 + K011 K200 + ^ H O ^ O l + K011 K100 + 2 K 1 1 0 K 1 0 0 K o o : L + SKioAooSlO + K200 K010 K001 + ^OO^OIO^OI - N X P x 2 p 2 P 5 G^ 90 11. E(X1X2X5X^) = K l x l l + K 1 1 1 0 K 0 0 0 1 + K 1 1 0 1 K 0 0 1 0 + K 1 0 1 1 K 0 1 0 0 + K0111 K1000 + K1100 K0011 + K1010 K0101 + K1001 K0110 + K1100 K0010 K0001 + K1010 K0100 K0001 + K1001 K0100 K0010 + K0110 K1000 K0001 + K0101 K1000 K0010 + ^ o i A o o o ^ o i o o + ̂ OOĈ OIOĈ OOIĈ OOOI Now i t i s an easy matter to f i n d the moments about the o r i g i n . Equations 1, 3> 6, and 11 need no change. The others need s l i g h t modification which i s done as follows - 2'. E ( X x 2 ) m E ( X x ( 2 ) ) + E(X x ) = NXPl { [N(X+l)-l]p x + l } = NXPX ( P XG X + 1) 4'. E ( X x 5 ) = E ( X 1 ^ ^ ) + 3 E ( X x ( 2 ) ) + E(X x ) = NX P l(p x 2G 2 + 3P XG X + 1) 5«. E ( X X 2 X 2 ) = E(X x^ 2)x 2) + E(X x X 2 ) = KXP XP 2(P XG 2 + Gx) 7'. E ( X X 4 ) = E ( X x ( 4 ) ) + 6 E ( X X ( 5 ) ) + 7E ( X X ^ 2 ^ ) + E(X x) - Nxp x( P l 5G 5 + 6 p x 2 G 2 + 7P xG x + 1) 8'. E ( X x 5 X 2 ) m E ( X x ^ 3 ) X 2 ) + 3E(X x( 2)x 2) + E(X x X 2 ) = NXPXP2 (P x 2& 5 + 3p xG 2 + Gx) 91 9*. E ( X 1 2 X 2 2 ) = E f X ^ 2 ^ 2 * ) + E{X^2\) + EiX^k^+EiX^) = N\p xp 2 [ p X P 2 G 5 + ( p X + P 2 ) G 2 + G x] 10'. E ( X x 2 X 2 X 3 ) = E ( X 1 ( 2 ^ X 2 X 5 ) + E ( X 1 X 2 X 3 ) - NXPTLP^-JO?^ + G 2 ) Formulas 1, 3* 6 and 11 together w i t h the primed formulas give expressions f o r the moments of order f o u r or l e s s . These, i n a s l i g h t l y g e n e r a l i z e d form are summarized i n (1-3.4). 92 APPENDIX 1C CALCULATION OP THE ENTRIES OP THE INFORMATION MATRIX " J " The c a l c u l a t i o n of I i s o u t l i n e d i n §1.5B. Consider I . and I . X P j P j X I,- = I . = E( l o g L l o g L ) toy d e f i n i t i o n . x p J p r ax apd D i f f e r e n t i a t i n g a f t e r s u b s t i t u t i n g from (1-4.1) y i e l d s 6, P 6 6 « y y E[ ' 1 i - p-(5?) . 1 • i - p-»(x )] A Y = l W ^ X a W 9 P J Y Because the observations are independent and i d e n t i c a l l y d i s t r i b u t e d I,„ = 0 E ( i £- P-(x) . -2- P->(x)) X P J U P - ( X ) ] 2 ax X ^ a P j x ^ * + 0(0-1) E [ — 3 - P^(x )] E [ - k - P^(x )] (1C-1) P-(X) ax x a P-(X) a P j x a The second term i s zero by the same reasoning as i n (1-5.8). From the d e f i n i t i o n of e x p e c t a t i o n , and s u b s t i t u t i n g from (1-4.4) and (1-4.7) f o r the d e r i v a t i v e s , 93 (i/e) i . = I... Y ( p " ( y 1 } P s (? +e m) n-1 x + ^ + (1/NX) £ ^ P ^ x ) - P^(x)) ( ( x , / P J . ) P - ( x ) - i l l - P^x+e^)] k=l P j As I s seen from (1-4.7) t h i s h olds f o r a l l m. I f we choose m=j, m u l t i p l y the expression out and replac e the sums by corres- ponding expectations and use lemma 1-5 where necessary, ( 1 / P ) l x p =-!-§— E f X ^ - l ) ] - p nNX(A+l)+(l/p JNX)EfX J) d P^j NX I f we choose m4j and f o l l o w the same procedure, (1/3) I x p = P n E ( X j X j - p nNx(A+l)+(l/p ; jNX)E(X j) J P^PjnNX I n e i t h e r case, we may s u b s t i t u t e f o r the expectations u s i n g ( l - 3 « 3 ) . Both w i l l l e a d t o the same r e s u l t which i s (1/p) I = - P rNXA + P nN + 1 - p n (1C-2) Now consider I . By d e f i n i t i o n I ^ = p i p d p i p j E[a/ap., ( l o g L ) . a/ap . ( l o g L ) ] . L e t us now s u b s t i t u t e from (1-4.1) and r e a l i z e t h a t the observations are independent and i d e n t i c a l l y d i s t r i b u t e d . Then p i p j [P^(X)]2 a P ± x a P j x + B(B-I) E M ~ ~ p5?( x)3 E [ — L - . P-(x)] P^(X) ap± x P-(X) a P j Reasoning as i n (1-5.8), we see tha t the second term i s zero. The s u b s t i t u t i o n f o r the d e r i v a t i v e s from (1-4.4) y i e l d s CO CO ^ x.+l [ ( ^ / P i ) P^(x) + - i — p l ( x + e . ) ] p i I f i = j , we use the d e f i n i t i o n of ex p e c t a t i o n and expand the expression. Then (1 /P) Ip p - ( l / p i 2 ) [ E ( X i 2 ) - 2 E ( X i ( X i - l ) ) + E2\%±2(A+1)] and the use of (1-3.3) and s i m p l i f i c a t i o n g i v e s (1/p) I p . N 2\ 2A + NX ( l / p ± + 1-N) (1G-3) I f ±4i> w e m a v u s e the same procedure t o get (1/P) 1 ^ = ( l / p i p J ) [ - E ( X i X j ) + N ^ S j P j C A + l ) ] = N 2X 2A +NX(1-N) (1C-4) Equations 1C-1, 2, 3> and 4 give the remaining e n t r i e s of the in f o r m a t i o n m a t r i x . 95 APPENDIX ID CALCULATION OP THE INVERSE OF THE MATRIX J/B = Q R R S+W/p1 R S • * R S Step 1. Fi n d det (J/B) R S S+W/p, R S S 3+W/p, n-1 Let us perform the f o l l o w i n g elementary row operations on J/p. 1. Subtract column 2 from columns 3, 4, n-1. 2. Subtract (S/R) (column 1) from column 2. 3- Add (pj^/p-j^) [row ( k + l ) ] t o row 2 f o r k=2, 3, n-1. 4. Add' (p.j/W)(QS/R-R) (row 2) to row 1. These operations leave the determinant unchanged, w i l l have the r e s u l t Hence we det (±) = B Q + — (— - R)(R + R V ^ ) W R .n p. n-1 Pj k=2 p l W/Pl n - 1 W/p, n-1 det (J/B) = [Q+(Q3-R2)(l-pn)/¥3"n" (W/p± ) 1=1 (1D-1) Step 2. F i n d minors of m a t r i x elements Let K be the minor of the (ct,y) entry of J/B. T K S + ¥/p x S + W/p, S S + W/p. n-1 Carry out the f o l l o w i n g elementary row o p e r a t i o n s . 1. Subtract column 1 from columns 23 J>, ..., n-1. 2. Add (Pfc/Pi) (row k ) t o row 1 f o r k=2, 3> n-1. The r e s u l t i s a lower t r i a n g u l a r m a t r i x whose determinant i s n-1 K u - [1 + S(l-p n)/W] ~ y j (W/p ±) (1D-2) i = l K X PJ R P l S + s w_ p i - 1 S + JL 97 Do the f o l l o w i n g o p e r a t i o n s . 1. Subtract row i from a l l other rows. 2. Expand by column 1. The r e s u l t i s a diagonal m a t r i x whose determinant i s column i+1 i ^ J 98 Do the f o l l o w i n g operations 1. Subtract row J from a l l other rows except row 1. 2. Subtract (R/S) (row J ) from row 1. 3. Expand by column (1+1) and then by row 1. K_ _ = ( - l ^ + ^ Q S - R 2 ) I \ (W/p ) P i P 0 K P*P Q R R S + | - p l S+ w 5 i - l R p i + i s+ w >n-l (1D-4) This w i l l be the same as det ( J/B) but w i t h the expressions i n p^ m i s s i n g . Thus K = [Q+(QS-R 2)(l-p B- P i)/W] J T (¥/pfc) p i p i k 4 i (1D-5) Prom e i t h e r (1-5.6) or (2-5.3) we know tha t f\ = J , i . e . fl = (1/B)(J/S )""*". By a w e l l known theorem i n m a t r i x theory, the elements of II can be expressed as the c o f a c t o r s T of corresponding elements of J d i v i d e d by det J . Hence from (1D-1), (1D-5), v a r x - 1 ^ - 1 W+s(l-p p) \ p d e t ( j / 0 ) p WQ+(QS-R^)(l-p n) r n v ,* - * 1 K p i X 1 R P j COV (X,p. ) = p- ~ ~ a — 5 — 1 p d e t f J / 0 ) P WQ+(QS-R-)(l-p n) cov ( M j ) = | . ̂  . - 1 1 3 P d e t ( J / 0 ) p [WQ+(QS-R*)(l-p n)]W var p. = J . P i P l = - I — ^ 1 1 P o ^ j T ^ P [WQ +(QS-R 2)(l-p n)]W (ID 100 APPENDIX IE Lemma; Le t m = (ou±, ..., i»n) and £i » (u^, ..., u^) be two ..sets of random v a r i a b l e s r e l a t e d by the equations UD^ = (^(M), i = 1, 2, ..., n. L e t and ~\x^ denote the expected values of u)x and r e s p e c t i v e l y , and A = (-H^) and M = ( M ^ ) denote the covariance m a t r i c e s of w and # r e s p e c t i v e l y . A l s o , denote by 0, the Jacobian A L i l = du)4 <M=W' Then i f higher order d e r i v a t i v e s of w i t h respect t o are n e g l i g a b l e compared t o the f i r s t order, we have f l = (}̂ M(|, Proof: Using T a y l o r ' s theorem, n k=l 9 ( ik UO=UD (^-1-^) + second order terms Thus n n Z v-i a«)-. _ _ k=l m=l 9 | Jk 9*m where the d e r i v a t i v e s are evaluated at uu n i t i o n s of fl and M, - w Prom the d e f i - ±3 n n = 1 I k=l m=l 9 l ik B|Jm M. - ( 4 T M(J) i d Thus each entry of . f l agrees w i t h each entry of <JT M(J. Q.E.D. By a w e l l known theorem from m a t r i x theory, the lemma i m p l i e s det fl = det ( J T . det M . det (J, o r det Pi = (det 4 ) 2 det M. APPENDIX I F CALCULATION OF (1-6.12) AND (2-6.5) Let us s u b s t i t u t e (1-3.3) and (1-3.4) i n t o (1-6.11), cov (W2,\) = ( l / 6 ) ^ P k N X ( p f e 2 G 2 + 3P kG x + 1) + 2 I N X P k P j ( P k G 2 - C l ) + T 1 ^ V k P i P j j=fk i 4 k d4k,i +A N X P A ( p i G 2 + G l } " W X P k I N X P i ( P i G l + X ) i 5 f k i = l n-1 " N X p k I I N\G l P ip. i = l j 4 i A f t e r c a r r y i n g out the summations we o b t a i n = ) { P kNX(p k 2G 2+3P kG 1+l )+2N\p k(p kG 2+G 1 )(1-P f c-P n ) + N X p k G 2 [ ( l - p k - p n ) 2 - I P ± 2]+NXP k[G 2 E P i 2 + G i ( l - P k - P n ) i 4 k ±4^ n-1 n-1 - N 2 X 2 p k ( G x I P i 2 4- l - P n ) - H 2 X 2 G l P k C ( l - p n ) 2 - I Pl2] i = l i = l Upon s i m p l i f i c a t i o n , t h i s y i e l d s cov (V2,\J = ( l / p ) N \ p k { ( l - P n ) [ G 2 ( l - p n ) + 3G X - N X < G 1 ( l - p n ) + l ) ] + l } This equation i s (1-6.12). APPENDIX 1G CALCULATION OP (1-6.16) AND (2-6.6) Let us s u b s t i t u t e (1-3-3) and (1-3.4) I n t o (1-6.15) _ n-1 w W2 - ( l / p ) ( I I I I ^ j P i P j V m 1-1 j 4 i k 4 i , j m 4 i 5 ^ k n-1 n-1 + 6I I I NXfGjPjSjPfc. + O ^ p ^ ) + 3 £ £ NX 1=1 J4I k 4 i 5 j i = i J4I n-1 [ G 3 P i 2 p / + r i 2 ( p i 2 p J + p i p j 2 ) + G l p i p ^ + * I I NX i = i J4I n-1 ( G 5 P i 5 P j + ^ G g P ^ j + C l p i p ^ + I Kx(G 3P i 2 r+6G 2P i 2+7G 1P i+p i) 1=1 n-1 n-1 2 .[ i N X P ^ P ^ + I H ^ r ^ w ^ " } 1=1 i = i d4i A f t e r c a r r y i n g out the summations we ob t a i n n-1 n-1 n-1 ^/^{m^ia-pj - 6 ( i - P r / l p k s + x l P k 2 ) 2 + 8 ( i - p n ) £ P l k=l k=l 1=1 n-1 n-1 n-1 n-1 -6 I p ± * ] + 6 f f X [ 0 5 < ( l - p n ) ^ P l 2 - 2 ( l - p n ) £ p / ) 2 1=1 1=1 1=1 J=l n-1 n-1 n-1 +2 I P i 4 ) * » 8 ( ( 1 - P n ) 5 - 3 ( l - P n ) I p / + 2 I p ±5 ) ] 1=1 'J=l i = l n-1 n-1 ( n-1 n-1 +3NX[G3<( ^ p . 2 ) 2 - 1 ^ ) ^ ( 2 ( 1 ^ ) lv*-2 lv±3) 1=1 1=1 'j=i 1=1 n-1 11' n-1 n-1 +0 1 < ( l - p n ) 2 - I p , 2 ) ] +4HX[G 3 ( ( l - p n ) £ P l ' - X Pl4> 1=1 1=1 1=1 i o 4 + 3 G 2 ( ( 1 - P n ) n i P ^ - T P I 3 > «i < ( l - P n ) 2 - ^ 2 > 1 i=i i = l n-1 1=1 n-1 n-1 n-1 +NX(G 3 I P i 4 + 6 G 2 I 4+7^1 P ^ + l - p J 1=1 1=1 1=1 n-1 n-1 ~ -[IU<G x £ p ± 2 + i - P a > +NXGx < ( i - P n ) 2 - £ P ± 2 ) ] 2 } i = i i = i Upon s i m p l i f i c a t i o n , t h i s y i e l d s var W2 = (l/0 ) N X(l-p n) { G ? ( l - p n ) 5 + 6 G 2 ( l - p n ) 2 + 7 G x ( l - p n ) + l - N x C l - p ^ t G ^ l - p ^ + l ] 2 ] This equation i s (1-6.16) 105 APPENDIX 1H CALCULATION OF det M WHERE M IS GIVEN BY R ( i - P n ) H 2 Rp 2H x R P n - l H l B P A Rp 1(p 1N»+l) RN'p 1P 2 RN' P lP 2 RN'p,p „ F l F n - l R P n - l H l RN 'Pl p n - 1 Let us perform the f o l l o w i n g elementary o p e r a t i o n s . 2. 3. Take the common f a c t o r R out of each row, the common f a c t o r H^ out of row 1 and column 1, and the common f a c t o r p^ out of row 1+1, i = l , 2, n-1. M u l t i p l y row 1 by -N' and add t o rows 2, J>» n. M u l t i p l y row i+1 by -p^ and add t o row 1, 1*1, 2, n-1. We now have n n-1 det M - (£.)" H12(TTP1) i=l n-1 = ( R / B ) n ( 7 f P i ) ( l - p n ) { H 2 - H 1 2 ( l - p n ) H 2 H ' ] 1=1 „ [1 ^ J ( l - P n ) H, 0 •. 106 APPENDIX 2A OBTAINING THE PROBABILITY GENERATING FUNCTION g*(s ) Let us s t a r t w i t h (2-2.5^"). This expression i s equal to ee x 2=o -X-,- — -Nz 5 -x„-...-Nz ) ( - S 2 P 2 ) 2 Z ( 2 x' ) ( " V l ) 1 + e -X x^=o 1 We can now apply (2-2.5) t o the l a s t sum and o b t a i n n-l z=l xn-l=° x 2=o -x,-.;.-Nz x c -x_-...-Nz , 6 0 -x-,-...-Nz - s 0 p 0 X 2 -x 0-...-Nz . x 2=o X, l - s l P l A p p l y i n g (2-2.5) again and co n t i n u i n g i n t h i s manner 00 £ (X ZA!)p n I , Z(l- S lP 1-...-VlP„-ir I , ZH- e" X z=l oo =e" X £ U z A ! ) p n N Z ( l - s 1 P 1 - . . . - s n _ 1 P n _ 1 ) -Nz z=o The sum i s simply the power s e r i e s expansion of N exp ( X / \ - X 1- £ S i P i * 1=1 from which (2-2.6) f o l l o w s 107 APPENDIX 2B CALCULATION OP THE ENTRIES OF THE INFORMATION MATRIX J * The c a l c u l a t i o n of 1 ^ * i s done I n §2.5B. Consider I . * and I *. By the same argument as i n xp.j PjA Appendix 1C, we may o b t a i n the same equation as (1C-1), and as i n (1-5.8), second term w i l l be zero, i . e . XV3 V6X 1 [ P x ( x ) ] 2 SX X 5Pj > S u b s t i t u t i o n of (2-4.2) and (2-4.3) i n t o the above equation r e s u l t s I n 1 x,=o x ,=o r x W t «*p m 1=1 w l " " ~ n - l " n-1 x. +1 4 - l l P ^ x ) ] • { [ x . / V d / p J ^ x k ] P , ( 5 c ) - - i — P x ( x + t i ) } C k=l p n p i (2B-1) From (2-4.3), we see t h i s must ho l d f o r any m, m=l, 2, n-1. I f we choose m=I, expand t h i s expression, and make use of lemma 2-4, we can write the above equation i n ex p e c t a t i o n n o t a t i o n as n-1 ( l / B ) l X p * = ( l / P i ) [ ( - l + l / P n ) E ( X 1 ) - ( l / N X ) E ( X i Y \ ) 1 k=l n-1 n-1 +(l/NXP n) ( ( 2 / P i ) E [ X i ( £ X k - 1 ) ] - E [ ( £ X k ) 2 ] ] k=l k=l n-1 - ( l / p n ) I E(X k ) - ( W x/p n)(A +l)+(l/NX)[E(X i 2)-E(X 1)] k=l S i m i l a r l y , i f we choose m=)&, we o b t a i n n-1 ( l / 6 ) l x p * = ( l / p 1 ) [ ( - l + l / p n ) E ( X 1 ) - ( l / K x ) E ( X 1 I X^) 1 k=l n-1 n-1 + ( l / H X p m ) f ( 2 / p m ) E [ X f f i ( I X k~l)]-E(£ X k ) 2 ) k=l k=l } n-1 -(l/p n)£ E ( X k ) - ( N X / p n ) ( A + l ) + ( l / N X P 1 P m ) E ( X 1 X m ) k=l Upon s i m p l i f i c a t i o n by means of s u b s t i t u t i o n from (2-3.3) of e i t h e r of the above expressions, we f i n d t h a t (1/P )lXPi*=NxA/pn+[ (NX+N+1 ) p n 2 - l / p n - NX]/pn (2B-2) Next consider I p . Prom (2-5-5) and same procedure as above, P i P J [ P ^ ( x ) j - BP ± X BX X Let us s u b s t i t u t e f o r the d e r i v a t i v e s from (2-4.2) and expand oo » n-1 1 J X t=O x n=o ^ x ^ ' k=l " P x ( x + ^ i ) } { ^ j / P j + C ^ P n ) I P x ( x ) p n p i k=l x +1 p n p j 109 I f I = j , expansion and the d e f i n i t i o n of exp e c t a t i o n y i e l d 2 ( 1 / P ) I P i p i * = ( 1 / P i ^ 1 " 2 / p n ) E ( X i M s / P n P i H d / P i n-1 n-1 + l / p n ) E ( X 1 ) + ( l - l / p n ) E ( X i £ X k ) + ( l / p n 2 ) E ( Y \ ) k=l k=l + ( N 2 X 2 / p n 2 ) ( A + l ) S u b s t i t u t i o n f o r the expectations u s i n g (2-3.3) w i l l g ive us (1/0)1 • = N 2 X 2 A / p n 2 + (NX/p n 2)[-(NX+N+l)/p n 2 + l / p n +NX+1] + NX/pNPI (2B-4) I f 14J, (2B-4) y i e l d s P i p / " ( 1 - 2 / p ^ L , , , ^ n-1 + ( i/ P j)E(x d Y \ ) i + ( ^ n 2 ) { e ( E \y k=l k=l n-1 n-1 - (i/p d)ECx d( Y V 1 ^ • (V P 1 ) E [ X 1( J X^.-!)] n-1 ( l / P ) l p , P , * - ( l - 2 / p n ) E ( X 1 X j ) + ( l / p n ) [ ( l / p 1 ) E ( X 1 I X ^ k=l n-1 n-1 k=l " k=l n-1 n-1 k=l k=l + ( N 2 X 2 / p n 2 ) ( A + l ) = N 2X 2A/p n 2+(HX/p n 2 )[-(NX+N+l)/P n 2+l/p n+NX+l] (2B-5) Equations 2B-1, 2, 4 and 5 give the remaining e n t r i e s of the i n f o r m a t i o n m a t r i x . l i p det (J* = APPENDIX 2C CALCULATION OP det (J* WHERE -P n Q p n p l XQ XQ Pn^Pn-1 XQ F "Q P l D -Pn XQ NX PgD XQ XQ P XD xo~ PoD n XQ NX 'rill XQ D F P XD XQ XQ P i D Fn-1 . XQ I E NX To f i n d the determinant we must perform the f o l l o w i n g elementary operations. 1. Fac t o r Q out of every row of the m a t r i x , p R out of column 1, and f i n a l l y p x/X out of row i+1 f o r i = l , 2̂  • • • £ x^"*l • 2. M u l t i p l y column 1 "by F and add t o the other columns. 3. Subtract row 2 from rows"3* 4, n. 4. M u l t i p l y column i+1 by P^/P^ and add t o column 2 f o r 5. Expand by row 1 and then by column 1. n r The r e s u l t I s a constant times the determinant of a diagonal m a t r i x , and so -Hp n _ 1 Q (NX) n n 112 APPENDIX 3A CALCULATION OF EFFICIENCY OF METHOD OF MOMENTS FOR THE POISSON-MOLTIVARIATE POISSON DISTRIBUTION Equation (3-^.7) f f . suggests two ways t o f i n d the e f f i c i e n c y . We may take the l i m i t of the e f f i c i e n c y of e i t h e r the Poisson M u l t i n o m i a l or the Poisson-Negative M u l t i n o m i a l d i s t r i - b u t i o n as N->•«>, i . e . we may take l i m i t s i n e i t h e r (1-6.24) or (2-6.13). Method 1. Take l i m i t i n g value of (1-6.24). A f t e r d i v i d i n g numerator and denominator by N and rearranging the expression s l i g h t l y , Use (3-3.1), (3-3.2), and (3-3-3) t o f i n d the l i m i t of the f i r s t * f a c t o r i n the denominator and (3-4-3) t o f i n d the l i m i t of the second. A l s o use the f a c t t h a t a = N(l-p ). Then 1 ' [ H 2 < N ( l - p n ) + p n > -H^] E f f s a This s i m p l i f i e s t o (3A-1) 113 Method 2. Take l i m i t i n g value of (2-6.13). A f t e r d i v i d i n g numerator and denominator hy N and rearr a n g i n g the terms s l i g h t l y , ( l + l / N ) 4 N 5 ( l - p n ) 5 E f f = l i m <! — * xx ' v "xx V~PP }~(\v"™S I p n ^ X B X X * + < B X X ^ B P P * / N ) - ( B X p V N ) O P n N ( l - P n ) l 1 [ (1+1/N)N(l-pn )+D+Pp nJ 2[(H 2*-H 1* 2 )p n+(l-p n)N(1+1/H)H 2*] ^ L e t us note t h a t from (2-6.10), l i m Fp n = l i m ( 2 ( l - p n ) [ N ( X + l ) + l ] p n + P n 2 ) = 2(X+l)a + l l i m D = l i m f-K ( l - p )(l+p ) - 1 - 2(-^L) XP_N(l-p ) + ( — ) X 2 N 2 ( l - p ) 2 + (-*-) X ( l - P n ) 2 ) = -2(x+l)a - 1 N+l n N+l J Hence we see tha t l i m (D+Fp ) = 0 I f we use t h i s f a c t and equations (3-3.1), (3-3.2), and (3-3-3) t o f i n d the l i m i t of the second f a c t o r i n the denominator and (3-4-3) t o f i n d the l i m i t of the l a s t f a c t o r ,3, E f f = c r y |1«[XA+<A\ (XA-1) - (-XA+1T > «].a' . 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