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Majorization: its extensions and the preservation theorems Cheng, Koon Wing 1977

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MAJORIZATION: ITS EXTENSIONS AND THE PRESERVATION THEOREMS by KOON WING CHENG B.So., University of Hong Kong, 197^ A THESIS SUBMITTED•IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF t MASTER OF SCIENCE . in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS and the INSTITUTE OF APPLIED MATHEMATICS AND STATISTICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1977 © Koon Wing Cheng, 1977 In presenting th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l ica t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of Mathematics The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date W t S L \ - i i -Abstract. This thesis deals mainly with the orderings induced by majorizatIon, the two weak majorizations and t h e i r associated i n e q u a l i t i e s . One of these weak majorizations has received some attention i n the l i t e r a t u r e . However, the other one, being dual to the former, i s t o t a l l y overlooked. Schur functions which preserve these orderings are shown to have applications i n s t a t i s t i c s . In Chapter 1, we discuss b r i e f l y the majorizations and the i r r e lations with Schur functions, bringing out the exten-sions to the' weak majorizations where possible. In Chapter 2, we generalize the ordinary majorizations to those parametrized by a vector p of positive components. We discuss the proper-t i e s of these new majorizations i n a d i r e c t i o n p a r a l l e l to that of ordinary majorizations. The Schur functions are likewise generalized. In Chapter 3, we discuss the stochastic extensions of majorizations and the preservation theorem of Schur con-vexity due to Proschan and Sethuraman (1977). With a preservation theorem on monotonicity, we study the stochastic extensions of the weak majorizations. Some i n e q u a l i t i e s a r i s i n g i n some multivariate d i s t r i b u t i o n s are found to be direct consequences of the two preservation theorems. F i n a l l y , we consider the unbiasedness of a certain class of tests of signif i c a n c e . - i i i -Table of Contents Page Abs t rac t i i Table of Contents i i i L i s t of F igu re i v Acknowledgement V Chapter 0 : No ta t i on and convent ions . .' 1 Chapter 1 : A. I n t r o d u c t i o n . 3 B. M a j o r i z a t i o n and weak m a j o r i z a t i o n s . 3 C. Schur f u n c t i o n s . 8 Chapter 2 : A. I n t r o d u c t i o n . 14 B. The 'p4jTiaj"drjr:za.fel;ona. 1 ia t? •&. 14 C. C h a r a c t e r i z a t i o n by m a t r i c e s . 19 D. Func t ions p r e s e r v i n g the p - m a j o r i z a t i o n s . 24 Chapter 3 : A. I n t r o d u c t i o n . 29 B. S t o c h a s t i c ve rs i ons of m a j o r i z a t i o n s . 29 C . P r e s e r v a t i o n of Schur convex i t y . 35 D. P r e s e r v a t i o n of mono ton ic i t y . 42 E. A p p l i c a t i o n to hypothes is t e s t i n g . 46 B i b l i o g r a p h y 48 - I V -L i s t of Figure Page Figure . An i l l u s t r a t i o n of. maj o r i z a t i o n i n . 13 - V -Acknowledgement I would l i k e t o thank my s u p e r v i s o r , Dr. A.W. M a r s h a l l f o r h i s c o n s t a n t h e l p and encouragement. A l s o , I am g r a t e f u l t o t h e U n i v e r s i t y of B r i t i s h Columbia f o r the f i n a n c i a l a s s i s t a n c e . F i n a l l y , I w i s h t o e x p r e s s my g r a t i t u d e t o Mrs. E v e l y n Yang f o r her t y p i n g of t h i s t h e s i s . - 1 -C h a p t e r 0 N o t a t i o n a n d C o n v e n t i o n s F o r e a s i e r r e a d i n g o f t h i s t h e s i s , t h e f o l l o w i n g n o t a t i o n and c o n v e n t i o n s a r e e m p l o y e d t h r o u g h o u t . T h o s e n o t l i s t e d b e l o w w i l l be s p e c i f i e d w i t h i n t h e c o n t e x t . (1) R e a l numbers and v e c t o r s a r e d e n o t e d by s m a l l l e t t e r s , u s u a l l y by u , v , x , y , z ; random v a r i a b l e s a n d v e c t o r s a r e d e n o t e d by c a p i t a l l e t t e r s , u s u a l l y by X,Y; v e c t o r s a r e u n d e r l i n e d w i t h c o m p o n e n t s i n d i c a t e d by s u b s c r i p t s . (2) U n i v a r i a t e f u n c t i o n s a r e u s u a l l y d e n o t e d by f , g , h and m u l t i v a r i a t e f u n c t i o n s a r e u s u a l l y d e n o t e d by F,G,H. D e n s i -t i e s f o r b o t h u n i v a r i a t e and m u l t i v a r i a t e d i s t r i b u t i o n s a r e d e n o t e d by <j> o r j „ • ] | > A , ) ^ i f p a r a m e t r i s e d by A_. (3) The d e c r e a s i n g o r d e r a r r a n g e m e n t X J - ^ - J > ... > x | - n j - i s a p e r m u t a t i o n o f t h e components x^,-. ...x o f x. R e v e r s e o r d e r s t a t i s t i c s X r . -, > ... > X r -, a r e d e f i n e d i n l i k e L U - - LnJ manner. (4) I f TT i s a p e r m u t a t i o n o f { l , . . . , n } , we d e n o t e by x t h e v e c t o r (x ,...,x ). V % (5) O r d i n a r y o r d e r i n g o f v e c t o r s a n d m a t r i c e s i s c o m p o n e n t w i s e . (6) M o n o t o n i c i t y r e f e r s t o b o t h u n i v a r i a t e and m u l t i v a r i a t e f u n c t i o n s , t h e l a t t e r w i t h r e s p e c t t o t h e o r d e r i n g d e f i n e d i n (5 ) . A l s o , by i n c r e a s i n g ( d e c r e a s i n g ) we s h a l l mean n o n - d e c r e a s i n g ( n ' d n - i n c r e a s i n g ) . - 2 -(7) The abbreviations " i f f " f o r " i f and only i f " and " i . i . d . " f o r "independent and i d e n t i c a l l y d i s t r i b u t e d " are used. (8) L i m i t s of i n t e g r a t i o n w i l l be from -« to °° unless otherwise s t a t e d . - 3 -Chapter 1 A. I n t r o d u c t i o n . I n t h i s c h a p t e r , a b r i e f survey of the concept o f m a j o r i z a -t i o n and Schur convex f u n c t i o n s i s d i s c u s s e d . But some p r o p e r t i e s o f m a j o r i z a t i o n w i l l be g e n e r a l i z e d i n Chapter 2 . The main theorem i n t r o d u c e d here g i v e s e q u i v a l e n t c o n d i t i o n s o f m a j o r i z a t i o n s . These c o n d i t i o n s w i l l be extended t o t h e i r s t o c h a s t i c v e r s i o n s i n Chapter 3 • B. M a j o r i z a t i o n and weak m a j o r i z a t i o n s . M a j o r i z a t i o n i s a p a r t i a l o r d e r i n g d e f i n e d f o r p a i r s o f v e c t o r s b oth l y i n g on the same h y p e r p l a n e { x : x-^ +:.:., •+x n = K } i n R . D e f i n i t i o n 1.1. x i s s a i d t o be m a j o r i z e d by y , i n n o t a t i o n 21 •< y > i f f k k [ i ] [ i ] f o r k = l , . . . , n - l (1.1) and [ 1 ] E i ] l = l 1=1 Note t h a t we have the f o l l o w i n g r e l a t i o n s : n n (1) the l a s t e q u a l i t y i n (1.1) i s the same as j x. = J y. i = l - i = l - 4 -(2) i f x •< Z a n c i Z 2L J t h e n x and y d i f f e r by a p e r m u t a t i o n of t h e i r components; (3) i f x -< y J t h e n x71" ^  y 7 7 f o r any p e r m u t a t i o n s TT and I f we extend the i n e q u a l i t i e s t o t h e l a s t e q u a l i t y as w e l l , t h e n we have one v e r s i o n o f weak m a j o r i z a t i o n . D e f i n i t i o n 1.2. x i s s a i d t o be weakly m a j o r i z e d by y , i n k k n o t a t i o n x ^ y , i f f J x r i l < ^ ^ f i l ^ o r . . . ,n . i = l i = l We can r e w r i t e the system (1.1) i n an o t h e r manner, as n n I x m > I y r n f o r k = 2 > - - - > n i=k i=k L n n and T h i s suggests., a n o t h e r e x t e n s i o n t o weak m a j o r i z a t i o n o f a second t y p e : n n D e f i n i t i o n 1. 3. x % y i f f j x [ J j > j f o r k = 1 n . i=k i=k ' ' Each o f the s e weak m a j o r i z a t i o n s , and , a l s o d e f i n e s a p a r t i a l o r d e r i n g on R n. U n l i k e m a j o r i z a t i o n , v e c t o r s r e l a t e d by t h e s e o r d e r i n g s need not be c o n f i n e d t o a h y p e r p l a n e . T r i v i a l r e s u l t s f o l l o w i n g from t h e s e d e f i n i t i o n s a r e : - 5 -P r o p o s i t i o n 1.4. (1) x -< y i f f -x -< -y; (2) 2L y i f f -x ^ -y. M a r s h a l l and O l k i n ( t o a p p e a r ) use (2) as t h e d e f i n i t i o n o f w, . But t h e p r o p e r t i e s o f ™< , b e i n g d u a l t o t h o s e o f s e l d o m a p p e a r i n t h e l i t e r a t u r e . Next we g i v e two r e l a t i o n s b e t w e e n m a j o r i z a t i o n and weak maj o r i z a t i o n s . P r o p o s i t i o n 1.5. (1) x (}*> ) y i f f t h e r e e x i s t s u s a t i s f y i n g x -< u and u < ( > ) y , (2) x ^ y i f f x ^ y and x *k y. P r o o f : (1) I f x "w* ZJ l e T J ^ ^ e t h e i n d e x f o r w h i c h yk i s t h e s m a l l e s t component o f y. Take d=(y^ + ... + yn) - ( x ^ + ... + x n ) > 0. L e t u be t h e v e c t o r o b t a i n e d f r o m y_ by r e p l a c i n g y ^ by y ^ - d . Then i t i s e a s y .to v e r i f y t h a t u s a t i s f i e s x -< u and u < y. The c o n v e r s e o f (1), and (2) a r e t r i v i a l . A l s o , x ^ Z l f f -2L ~y l f f ~* -< ""^J " I i < -Z. f o r some u i f f x ^ v ( =-u), v > y f o r some v. We can a l s o show t h a t x v£ y i f f t h e r e e x i s t s u s a t i s f y i n g x < u and u .< y. But i n t h i s c a s e , we o b t a i n u^ by a d d i n g d^ t o x^. The p r o o f i s more i n v o l v e d b e c a u s e o f t h e p r e s e n c e o f t h e i n e q u a l i t y c o n s t r a i n t s n ot t o be v i o l a t e d . - 6 -D e f i n i t i o n " 1.6. A matrix with non-negative e n t r i e s i s c a l l e d doubly s t o c h a s t i c (doubly sub s t o cha s't i c ) i f each row sum and column sum i s equal to ( l e s s than or equal to) 1. The f o l l o w i n g theorem c h a r a c t e r i z e s , and i n terms of doubly s t o c h a s t i c and substochastic matrices. Theorem 1.7- (1) x y i f f there e x i s t s a doubly s t o c h a s t i c matrix P such that x = yP. (2) x y i n R^ (x -<Jj y i n R^) i f f there e x i s t s a doubly substochastic matrix. P such that x = y P, where R +(R_) i s the set of non-negative (non-positive) numbers. The ppoof i s omitted here because t h i s theorem i s not r e l a t e d to the f o l l o w i n g d i s c u s s i o n . Those i n t e r e s t e d are r e f e r r e d to Mar s h a l l and O l k i n (to appear). The next theorem i s a powerful t o o l i n m a j o r i z a t i o n , as i t n 2 reduces many proofs concerning m a j o r i z a t i o n from R to R . 1 h Theorem 1.8. x ^ y i f f there e x i s t u , ..., u such that x - - ] i ^ ^ ^ ' - > " < y . - ^ i L = y where f o r 1 = 0, 1, . . . , h, u 1 and u1+"'" d i f f e r i n only two components. Proof: Notice that f o r any vector z_, z, and 'z_V = (z[]_]> •••> z [ n ] ^ d i f f e r by a permutation which can be represented as products of t r a n s p o s i t i o n s . So we. can assume that x^ > ... > x , and V± > ••• - y n ' Now we proceed by i n d u c t i o n on n . I f any of the i n e q u a l i t i e s i n (1.1) i s a c t u a l l y an e q u a l i t y , then f o r some k , - 7 -x.a = ( x x , . . . , x k) < ( y ] _ J . • • ,y k) = y a , x^ = (x, ,x ) ^ (y. ,y ) = y b . — k+1' 5 n ^ J k + 1 5 5 J n — By the i n d u c t i o n a s s u m p t i o n , x •< v ^ ... ^ v .< Z. a n < ^ x < w •«<•••«< w -<y , so x = ( x , x ) - < ( v , x ) - < . . . ^ ( y _ r , x b ) < ( y a , x b ) -< ( y 3 - , ^ 1 ) -< . . . -<(y_a,ws) -< (£.a,y_b) = y_ , where two a d j a c e n t v e c t o r s d i f f e r i n o n l y two components. I f a l l t h e i n e q u a l i t i e s a r e s t r i c t i n e q u a l i t i e s , we t a k e 6 = min { (y +. . . + y f c ) - ( x . ^ . . . +x f ci) } > 0 k = l , . . . n - l and x = (x., +6 ,x„, . . . ,x . ,x -6) . — 1 ' 2 ' ' n-1 J n Then x ^ x and we can a p p l y the p r e c e d i n g arguments t o 6 6 x and y_ , because x ^ y and a t l e a s t one o f the i n e q u a l i t i e s of t h e i r components i s an e q u a l i t y . I t i s i n t e r e s t i n g t o note t h a t i f x and y_ a r e s i m i l a r l y o r d e r e d , t h e n we can choose the u 1 i n such a way t h a t they are a l l o r d e r e d i n the same manner as x and y . The p r o o f p r e s e n t e d here i s d i f f e r e n t from the one o r i g i n a l -l y g i v e n by Hardy, L i t t l e w o o d and P 6 l y a ( 1 9 5 2 ) , who c o n s i d e r e d the d i s c r e p a n c i e s (number o f d i f f e r e n t components) between x and . But i t w i l l be seen i n Chapter 2^: t h a t t h e above p r o o f can be employed t o prove', a more g e n e r a l r e s u l t . i i+1 Because u -< u and they d i f f e r i n o n l y two components, we can r e l a t e them by a doubly s t o c h a s t i c m a t r i x as : - 8 -1 i i+1 a a u = u a a •1 for s'omeaa.a s a t i s f y i n g a>0, a>0 and a+a=l. A matrix of this form, with 1 i n the diagonal entries except the two a , and 0 in a l l the other entries except the two a , corresponds. .%o a -"T-transform". The main Idea of Theorem 1.8 i s that: I f x -< y , then x can be derived from y by a f i n i t e number of T-transforms. C. Schur functions. Now we consider functions that preserve majorization. n D e f i n i t i o n 1.9- Let S be an ar b i t r a r y subset of R . A function F:R (S)* >• R i s said.to be a Schur convex function (on S) i f f for any x, y (eS), x - < y = ^ - F ( x ) < F(y), and F i s Schur-concave i f f -F i s Schur convex. It follows from this d e f i n i t i o n that a Schur convex (or Schur concave) function (on S) i s necessarily symmetric, because i f x' i s obtained from x by a permutation of i t s components (and i f x,x' e S), then x x 1 -< x ,implying F (x ) <F (x ' ) <F (x ) , so that F(x) = F(x') . The extension to weak majorization is the following: Proposition 1.10. F i s Schur convex and increasing (decreasing) i f f for any x, y, x ^ ( X () y = ^ F ( x ) < F(y) . - 9 -Proof : Necessity. Suppose that P i s Schur convex and increas-ing. Then for x ^  y , there exists u s a t i s f y i n g x .< u, u < y. So i t follows that F(x) < F(u) < F(y) . For s u f f i c i e n c y , notice that x ^  y whenever x -< y_ or x < y , Hence F(x) < F(y) whenever x ^ y of* x < y_ s Therefore F .'.-is Schur convex and increasing. The proof for the other version of weak majorization i s si m i l a r . The following theorem characterizes d i f f e r e n t i a b l e Schur functions i n terms of t h e i r derivatives. Theorem 1.11 (Ostrowski, 1952). A d i f f e r e n t i a b l e function F is Schur convex i f f It s a t i s f i e s : (1) F i s symmetric, (2) for a l l . i , j , (x.-x.)(|f-~- ||-) > 0 . The proof i n a s l i g h t l y d i f f e r e n t setting w i l l be given i n Chapter 2 . This theorem i s important i n i d e n t i f y i n g Schur convex functions because conditions (1) and (2) are usually easier to verigy than the direct d e f i n i t i o n i n the case of d i f f e r e n t i a b l e functions. It should be noted that the symmetry property (1) i s necessary as can be i l l u s t r a t e d by the following example. Example 1.12. Let F:R 2—*-R be defined by F ( x 1 } x 2 ) = (x 1-x 2) i f x-j^  > x 2 = 0 otherwise , - 10 -t h e n F I s c l i f f e r e n t i a b l e w i t h d e r i v a t i v e s s a t i s f y i n g ( 2 ) . But F i s not Schur convex because (3/4,1/4) -< (0,1) , but F(0,1) = 0 < 1/4 = F(3/4,1/4) . n C o r o l l a r y 1.13. I f f i s d i f f e r e n t i a b l e , t h e n F ( x ) = J f ( x . ) 1=1 1 i s Schur convex i f f f i s convex. A r e l a t i o n between Schur convex f u n c t i o n s and c o n v e x i t y i s : P r o p o s i t i o n 1.14. I f F i s symmetric and convex, t h e n F i s Schur convex. P r o o f : By Theorem 1.8, we can reduce the p r o o f to R . So l e t x l - x 2 ' ^1 - ^2 ' ^ ^ e r e m a r k f o l l o w i n g Theorem 1.8, x -< y = £ • ( x 1 3 x 2 0 = X ( y 1 , y 2 ) + A ( y 2 , y 1 ) f o r some X,X > 0, X + X = l . Hence F U - ^ x ^ < X P ( y 1 , y 2 ) + XF(y 2 , y ^ ) = X F ( y ; L , y 2 ) + X F ( y ; L , y 2 ) = P ( y 1 , y 2 ) . n P r o p o s i t i o n 1.15. I f f i s convex, t h e n F ( x ) = I f ( x . ) 1=1 1 i s symmetric and convex. The p r o o f i s s i m p l e and i s o m i t t e d . Now we s t a t e the theorem which w i l l be extended t o the s t o c h a s t i c v e r s i o n s i n Chapter 3 • Theorem I . l 6 . The f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t : (1) x -< y , n. (2) F ( x ) < F ( y ) f o r every Schur convex f u n c t i o n F on R , - 11 -(3) F ( x ) < F ( y ) f o r every symmetric,convex f u n c t i o n F on R , n n (4) £ f ( x . ) < £ f ( y . ) f o r every convex f u n c t i o n f . i = l 1 i = l 1 P r o o f : ( l ) = £ - ( 2 ) , (2)=^(3), (3)=^(4) f o l l o w r e s p e c t i v e l y from D e f i n i t i o n 1.9, P r o p o s i t i o n s 1.14 and 1.15-For (4 ) y*- (1), we may assume x.^v> > ... > x n and v l - ^2 - "•" - y n " F i F s t n o t i c e t h a t t h e f u n c t i o n s f ( x ) = x and f ( x ) = -x a r e convex, p u t t i n g them i n (4) y i e l d n n n n n n I x. < I y. and - £ x < - £ y , so t h a t £ x = £ y . i = l i = l i = l x z-i = l i = l i = l Next we see t h a t the f u n c t i o n s f ( x ) = ( x - y k ) + = max{0,x-y k} , k=l, . . . , n are convex, t h e r e f o r e k n x± + ... + x k - S i k y k = I f ( x ± ) < I f ( x 0 i = l i = l n k < I f ( y ± ) = I f ( y ± ) = y x + + y k - k y k , 1=1 i = l so t h a t Hence x ^  y k k . I x. < £ y. f o r k=l,...n-1 1=1 ~ 1=1 7V'; The e q u i v a l e n c e o f (1) and (4) i s proved by Hardy, L i t t l e -wood and P o l y a (1929), and t h a t of (1) and (3) i s proved by M i r s k y (1959). The e x t e n s i o n o f t h i s theorem to t h e two v e r s i o n s o f weak m a j o r i z a t i o n i s d i r e c t and s i m p l e . - 12 -Theorem I.17. The f o l l o w i n g a r e e q u i v a l e n t : (2) (1) f u n c t i o n F (3) F ( x ) < P(y) f o r every s ymmetric, convex and i n c r e a s i n g ( d e c r e a s i n g ) f u n c t i o n F , n n I f ( x . ) < ^ f ( y . ) f o r every convex and i n c r e a s i n g (4) ( d e c r e a s i n g ) f u n c t i o n f . The p r o o f i s c o n t a i n e d i n the p r e v i o u s d i s c u s s i o n and i s not r e p e a t e d . The e q u i v a l e n c e o f (1) and (4) i s proved by Tomic (1949). 3 F i n a l l y a g e o m e t r i c p i c t u r e o f m a j o r i z a t i o n i n R i s i l l u s t r a t e d . I n g e n e r a l , f o r a v e c t o r z e R w i t h d i f f e r e n t components, t h e r e are s i x p e r m u t a t i o n s o f t h e components. These s i x v e c t o r s a r e r e p r e s e n t e d by t h e p o i n t s A,B,C,D,E and F i n t h e f i g u r e shown below. They a l l l i e on the p l a n e x l + x 2 + x 3 = K • I f we l e t S be the s e t o f v e c t o r s m a j o r i z e d by z, and T the s e t o f v e c t o r s m a j o r i z i n g z_, t h e n i n the f i g u r e , S i s the convex hexagon ABCDEF, and T i s t h a t p a r t o f the p l a n e o u t s i d e the d o t t e d l i n e s . F urthermore S = {x:x -< z) ='{x:F(x) < F(z_) f o r every Schur convex F} , T =• {x:x >• _z} = ( x : p ( x ) > p(z_) f o r every Schur convex F} . I f 1 A i s t h e i n d i c a t o r f u n c t i o n o f the s e t A d e f i n e d by - 13 -l / v ( x ) = . 1 i f x e A 0 i f x i A , t h e n i t i s easy t o v e r i f y t h a t l g i s Schur concave and 1 T Schur convex. For any Schur convex f u n c t i o n F r e s t r i c t e d t o t h e p l a n e x l + x 2 + x 3 = K J F i n c r e a s e s a l o n g any r a y on the p l a n e from the c e n t r e (K/3 ,K/3,K/3). F i g u r e . An i l l u s t r a t i o n o f m a j o r i z a t i o n i n R . - 14 -Chapter 2 A. I n t r o d u c t i o n . I n t h i s c h a p t e r , we g e n e r a l i z e m a j o r i z a t i o n and weak m a j o r i -z a t i o n s t o the p - m a j o r i z a t i o n s p a r a m e t r i z e d by a v e c t o r p o f p o s i t i v e components. The H a r d y - L i t t l e w o o d - P o l y a i n e q u a l i t y i s extended t o a more g e n e r a l form. The p r o p e r t i e s o f the p - m a j o r l -z a t i o n s a r e d i s c u s s e d , w i t h the c h a r a c t e r i z a t i o n . b y a c l a s s of m a t r i c e s s i m i l a r t o t h e doubly s t o c h a s t i c m a t r i c e s . F u n c t i o n s p r e s e r v i n g t h e s e o r d e r i n g s a re c o n s i d e r e d . Throughout t h i s c h a p t e r , the parameter p i s a v e c t o r w i t h p o s i t i v e components. We s h a l l adopt the f o l l o w i n g n o t a t i o n : i f TT i s a p e r m u t a t i o n o f {1, . . . , n} , x e D means t h a t x > . . . > x : i n o t h e r words , the v e c t o r x 7 7 o b t a i n e d from p e r m u t i n g t h e components o f x under TT i s i n d e c r e a s i n g o r d e r . Moreover, we denote by x e D i f TT I s the i d e n t i t y mapping. B. The p - m a j o r i z a t i o n s . The f o l l o w i n g i s a g e n e r a l i z a t i o n o f m a j o r i z a t i o n . D e f i n i t i o n 2.1. x ^ y on D77 i f f x, y_ e D ^ C t h i s means t h a t t h e components o f x and y_ are s i m i l a r l y o r d e r e d ) , k k f o r k = l , n - l - 15 -n n n n and I P x I p y (or e q u i v a l e n t l y £ p,x,= £ p.y. ) 1 = 1 1 71 711 1 = 1 ^1 ^1 1 = 1 1 1 1 = 1 1 1 The e x t e n s i o n s t o x y and x y are o b t a i n e d by r e -k k p l a c i n g the above system by y y and i = l l l i = l I l n n T p x > J p y f o r k = l , ..., n. . , TT . TT. — 7 T . T T . i=k l l i=k I l One remark about t h i s p - m a j o r i z a t i o n i s t h a t v e c t o r s r e l a t e d by t h i s o r d e r i n g have t o by s i m i l a r l y o r d e r e d . Indeed, the o r d e r i n g depends on t h e p e r m u t a t i o n TT . The r i g o r o u s way o f w r i t i n g s h o u l d be x TT-^D y. However, the r e q u i r e m e n t x, y e D has made the dependence on cTrlce^ea^rncan^u Theorem 2.2 below i s an e x t e n s i o n of the H a r d y - L i t t l e w o o d -P o l y a i n e q u a l i t y w hich c o r r e s p o n d s t o t h e case I n which a l l the p^ a re e q u a l . n n Theorem 2.2. I f £ p . f ( x . ) < £ p . f ( y . ) f o r every convex f , TT 1=1 1=1 t h e n f o r any such t h a t x_, y_ th e n f o r any TT such t h a t x, y e D , we have x ^ y on D C o n v e r s e l y , i f f o r some TT , x ^ y,on D77 , t h e n n n J p . f ( x . ) < y p.'f(y.) f o r any convex f . . o -, 1 1 — . - , 1 1 1=1 1=1 P r o o f : We o n l y prove the case t h a t TT i s the i d e n t i t y mapping For the p r o o f o f the g e n e r a l c a s e , j u s t r e p l a c e a l l t h e s u b s c r i p t s i by . F i r s t suppose t h a t x, y e D and - 16 -n n I p ^ f ( x i ) - < I p . f ( y . ) f o r any convex f . Choosing f ( x ) = x and i = l >?i=l n n f ( x ) = - x , we get I p.x. = J p.y. • Next c h o o s i n g f ( x ) = (x-y, ) 1=1 1=1 f o r k = l , ..., n - 1 , we get J P i x i " J P i ^ k < I P i f ( x i } < j P i f ^ i > i = l i = l i = l i = l n k k k < I v±r(y±) = I P ± f ( y ± ) = I P ^ i - I P i y k . 1=1 1=1 1=1 1=1 k k so J p . x . < J p.y. i = l 1=1 C o n v e r s e l y , suppose x A> y on D. We may assume t h a t x f ^ y , f o r k k k k = l , n. D e f i n e A Q = 0, BQ=0, A k= £ p x ± , i = l k ^ . ^ P ^ i a r i d Q k = [ f ( x k ) _ f ( y k ) ] 7 [ x k " Y k ] f ° r k = 1 » n-Because f i s convex, we have Q k > Q k + 1-Thus T(Ak-Bk)(Qk-Qk+1) + ( A n - B n ) Q n < 0 , k=l t h e r e f o r e " f A k ( Q k - Q k + 1 ) + A n Q n < J ^ V W + V n , k=l k =l t h e r e f o r e J^\~\zV^ K- V B k - l ) Q k • so t h a t j ^ k ^ ^ k ^ ^ y k ) ^ ^ ^ ^ - ^ - ! ) - ^ ^ - ! ^ ^ < 0 ' I t can be seen from the p r o o f t h a t the converse i s t r u e f o r - 17 -any T h i s p a r t i s proved by Fuchs (1947). But the f i r s t p a r t o f the p r o o f r e q u i r e s t h a t a l l t h e p i a r e p o s i t i v e ( o r n e g a t i v e ) . However, t h i s c o n d i t i o n i s l e f t out by M i t r o n o v i c (1970), who quoted the s u f f i c i e n t c o n d i t i o n o f Fuchs as n e c e s s a r y and s u f f i c i e n t . I t can be proved t h a t f o r n=2, t h e theorem h o l d s f o r a r b i t r a r y p^ and p 2 > Yet from t h e f o l l o w i n g example we see t h a t i t does not h o l d i n g e n e r a l i f the p^ are o f d i f f e r e n t s i g n . Example 2.3- Take p 1 = p 2 = l , p'^=-l, y 1=x 1=x 2=x^ > y 2 = y t h e n x, y eeD ; and f o r any convex f , p-^f (x-^ ) + p 2 f ( x 2 ) + p ^ f ( x ^ ) < p±f(y1)+P2f(y2)+P3f^3)> b u t P 1 x 1 + p 2 x 2 > p 1 y 1 + P 2 y 2 . F i n a l l y , we remark t h a t i f x and y are not s i m i l a r l y o r d e r e d , t h e n the i n e q u a l i t y v w „ v w \ can e i t h e r H I P ± f ( x ± ) < 2 P ± f ( y ± ) c a i = l i = l be t r u e o r f a l s e . Take f o r i n s t a n c e , f(5)+f(3)+2f(1) < f(6)+ f(0)+2f(2) f o r any convex f because (5,3,1,1) -<(6,2,2,0). But f(5)+f(5)+2f(1) < ( o r >) f(6)+f(0)+2f(2) f o r any convex f i s f a l s e because (5,5,1,1) and (6,3,3,0) cannot be o r d e r e d by-<. J u s t l i k e t h e H a r d y - L i t t l e w o o d - P o l y a i n e q u a l i t y , i t i s p o s s i b l e t o ext e n d theorem 2.2 to convex f u n c t i o n s t h a t a r e i n c r e a s i n g (or d e c r e a s i n g ) by r e p l a c i n g >^ by -<Jp ( o r - ^ p ) . The o r d e r i n g i s q u i t e s i m i l a r t o m a j o r i z a t i o n . I t can be d e r i v e d by a f i n i t e number o f t r a n s f o r m a t i o n s s i m i l a r t o T - t r a n s f o r m s . A l s o , i t can be c h a r a c t e r i z e d by m a t r i c e s s i m i l a r - 18 -t o the doubly s t o c h a s t i c m a t r i c e s . Because o f the p r o p e r t y t h a t x y on D77 i f f x 7 7 ^  y 7 7 on D, we o n l y have to c o n s i d e r x y on D and t h e n extend t o D77 i n t he obVious way. Theorem 2.4. I f x ^ y on D , t h e n t h e r e e x i s t u 1 , . . . , u h such t h a t x E u° -£> u 1 . . . u h ^ = y on D, where f o r i = 0, 1, ..., h, u 1 and u1+"'~ d i f f e r i n o n l y two components. P r o o f : By I n d u c t i o n on n . We c o n s i d e r the f o l l o w i n g two cases : k k (1) I f any o f the i n e q u a l i t i e s J p.x. < J p.y. , k = l , i = l 1 i = l 1 1 n - l i s a c t u a l l y an e q u a l i t y , t h e n we can w r i t e x y on D as : a /a a „ b /b b ~ x y on D, x . <) y on D , , , a b s , a b N , a b N where x = (x , x ), y = (y , y ), P = (p , P ) • By i n d u c t i o n a s s u m p t i o n , t h e r e e x i s t v 1 , i = l , r and w^, j = l , ..., s such t h a t a ,& 1 /a /a r /a a ~ x < y _ < . . . - £ v - e £ y o n D , b /b 1 ,b y-b s ,b b _ x -4? w ^ . . . - 4 } w ^ y o n D , and such t h a t each two a d j a c e n t v e c t o r s above d i f f e r I n o n l y two components. Combining t h e s e v e c t o r s , we get x = ( x a , x b ) 4 ( v 1 , ^ ) 4 ... 4 ( f , x b ) 4 ( y a , x b ) -4 ( y a J w . 1 ) -4 • • • ^ ( y a , w s ) .4 ( y a 3 y b ) = y on D. I t i s easy t o v e r i f y t h a t a l l t h e s e v e c t o r s a r e i n d e c r e a s i n g o r d e r and t h a t any two a d j a c e n t v e c t o r s d i f f e r i n o n l y two - 19 -components. (2) I f a l l i n e q u a l i t i e s a r e s t r i c t , t a k e k 6 = . -, m i n ,{ I p. ( y . - x . )} > 0 k = l , n-1 ^^^I^I i / J and x 6 = (x 1+ i6/p 1, x 2 , ^ n _ 1 , x n " c V p n ) • Then x -4} x y on D . For the n-1 component I n e q u a l i t i e s o f x ^ y, a t l e a s t one must be an I n e q u a l i t y . Now a p p l i c a t i o n o f (1) t o x and y completes the p r o o f . C. C h a r a c t e r i z a t i o n by m a t r i c e s . L e t «Ap be a s e t of nxn m a t r i c e s d e f i n e d by Ap = {A > 0: f o r any y e D, yA e D and yA -4} y on D} The f o l l o w i n g theorem c h a r a c t e r i z e s the m a t r i c e s A o f .j4fp. Theorem 2.5- I f A > 0, t h e n A e-«4p i f f A s a t i s f i e s : (1) eA=e, where e = ( l , 1); i . e . columns o f A sum t o 1 , k (2) £ a., i s d e c r e a s i n g i n i f o r k = l , n-1, (3) A P t = P _ t 3 where p^ i s the t r a n s p o s e o f p. P r o o f : N e c e s s i t y . e_, -e e D =±^eA ^ e, -eA ^ -e on D ==^ > eA = e_. For k = l , n-1, l e t z_k be the v e c t o r w i t h the f i r s t k components e q u a l t o 1 and the r e s t e q u a l t o 0 . Then z k e D „.k ^ p k k k -•• :i 7, J a. . i s d e c r e a s i n g i n i . F i n a l l y , z_ A ^ z_ on D j = l 1 J k t k t k n k D — ^ z_ Ap = z p — ^ [ a. .p. = J p. f o r k = l , n. But 1=1 j = l 1 J J 1=1 1 - 20 -t h i s s e t o f e q u a l i t i e s i s e q u i v a l e n t t o Ap^=jj^. S u f f i c i e n c y . F o r y e D, l e t x=yA. For i n d i c e s h<k, V x k = j / h j ^ j - j / k j ^ j n - l m m m m T h i s i m p l i e s x e D. A l s o , t t t n n yp = xAp = x p - | P ± x ± = £pp ±y ± . 1=1 i = l For k = l , n - l , a p p l y i n g c o n d i t i o n s (1), (2) and (3), we o b t a i n k k n y P.x. = y y y . a . . p . =i J 3 i = i 1=1 1 1 J 3 k k k nn = I I y a p + I I y a p j = l 1=1 j = l l = k + l J J k k k n - I I y i a i i P j + 2 I y k a i i P j j = l 1=1 J J j = l i = k + l K 1 J 3 k k k n = I I ( y i - y k ) a i i p i + ^ ^ a i i p i j = l 1=1 1 K 1 J 3 k j = l 1=1 1 J J k k Ic = 1 1 ^ y i = y k ) a 1 1 P 1 + y k I p i j = l 1=1 1 K 1 J J j = l J k k k n = I I ( y i - y k ) a i i p i + y k I I a i i p i j = l 1=1 K i = l j = l 1 J 3 k k k n = I I y i a 1 1 P 1 + 1 I y k a i 1 P 1 1=1 J = l J J 1=1 j = k + l K 1 J J - 21 -k k 1=1 j = i L y i a i j - j k n I I 1=1 j=k+l y. a. . p . k n k = X X y i a i . i p i = I p i y i n I 1=1 j = l ^ J u 1=1 Hence x y on D . From the d e f i n i t i o n of *<4rp and the e q u i v a l e n t c o n d i t i o n s g i v e n i n the above theorem, we see t h a t > 4 p p o s s e s s the f o l l o w i n g p r o p e r t i e s : (1) p i s c l o s e d under m u l t i p l i c a t i o n . (2) A' £ I 0 1 A' 0 I 2^ where p' i s the v e c t o r o b t a i n e d from p_ by t r u n c a t i n g some l e f t , and/or r i g h t end components, 1-^,1^ a r e i d e n t i t y m a t r i c e s o f a p p r o p i a t e d i m e n s i o n s . (3) I f x a Z.a > X0 y b o n D a r e such t h a t x a = y a A a , x b = y b A b where A a e , A b e Av° , t h e n (4) / a b N / a b v (x ,x ) = (y ,y ) At A o • A a 0 1 and >-. £ 0 A b 0^ A b J rp i s convex, as can be v e r i f i e d u s i n g p r o p e r t i e s (1) to (3) of Theorem 2.5 • However, the extreme p o i n t s t u r n out t o be q u i t e c o m p l i c a t e d , u n l i k e the case o f doubly s t o c h a s t i c m a t r i c e s where the extreme p o i n t s a re s i m p l y the p e r m u t a t i o n m a t r i c e s . -. 22 -Now we come t o the c h a r a c t e r i z a t i o n o f by m a t r i c e s A o f Theorem 2.6. I f x, y_ e D, t h e n x y on D i f f t h e r e e x i s t s A ej&p such t h a t x=yA. For t h e p r o o f we need the f o l l o w i n g lemmas. r Lemma 2.7. Suppose l<r<n, s>t, y>0, and l e t a= J p., i = l 1 i = r + l T T = J - [ ( s - t ) + Y ( ^ + | ) ] 1 , t h e n ( l ) (s j ... } Sj t , ... , t ) — ( s ' j s ' 3 t aand I — " ; | e Av , , f ) A l l A12 A A 21 22 A A 21 22;; where A^^ i s r x r , A-^2, A2^, A 2 2 h a v e a p p r o p r i a t e d i m e n s i o n s ; each composed o f i d e n t i c a l columns, w i t h e n t r i e s g i v e n by: t h l k 1 e n t r y o f A 1 1 i s p.(l-£)/a t h i k 2 e n t r y o f A^ 2 i s p^n/a t h j k ^ e n t r y o f A 2 1 i s P j ? / b t h j k 2 e n t r y o f A 2 2 i s p . ( l - n ) / b f o r 1=1, f o r j =r+l ,...,n ; n (2) f o r c=_^p., (s+£, . . . ,s+£) = ( S+-X 3s 3. . . } S ) A 5 where A , composed o f i d e n t i c a l columns, w i t h p./c as t h e - 23 -i k t h e n t r y , s a t i s f i e s A e>4p. The p r o o f o f t h i s lemma i s j u s t a d i r e c t a l g e b r a i c v e r i f i c a t i o n and i s o m i t t e d . The i d e a i s t h a t we c a n d i s t r i b u t e some p o s i -t i v e q u a n t i t y f r o m t h e s m a l l e r t o components t o t h e l a r g e r s components e v e n l y , o r w h o l l y t o t h e f i r s t component, by some s o r t o f t r a n s f o r m a t i o n i n v o l v i n g t h e m u l t i p l i c a t i o n by an elem e n t o f Jo^ . ^ p Lemma 2.8. I f x e D, t h e n f o r any 6>0, t h e r e e x i s t s A e s u c h t h a t x=x^A f o r x^ = (x, + — , x„ , . . . ,x -, ,x — — ) . — — — 1 p , ' 2 5 5 n - l ' n p *± pn P r o o f : P a r t i t i o n x ] _ > • • • > x n _ i i n t o k g r o u p s , e a c h h a v i n g e q u a l components and s u c h t h a t components i n d i f f e r e n t g r o u p s a r e d i f f e r e n t , i . e . x= ( u ^ ,. . . ,u_k , x n ) . Now d i s t r i b u t e a p o s i t i v e q u a n t i t y y f r o m x n t o t h e components o f u^.; t h e n t o t h e com-p o n e n t s o f u.k_-]_; and so on u n t i l i t i s d i s t r i b u t e d t o t h e components o f u^, by r e p e a t e d a p p l i c a t i o n s o f (1) o f Lemma 2.7; where y i s c h o s e n s m a l l enough so t h a t i t does not e x c e e d 6 and t h e new v e c t o r s f o r m e d a l l b e l o n g t o D . F i n a l l y , a p p l y (2) o f Lemma 2.7 t o d i s t r i b u t e i t w h o l l y t o t h e f i r s t component x-j^. I f nrY<6< (m+1 )y j p e r f o r m t h e above p r o c e d u r e m t i m e s u s i n g y J t h e n r e p e a t i t a g a i n u s i n g 6-my i n p l a c e o f Y • Be c a u s e o f p r o p e r t i e s (1) and (2) o f t h e f i n a l m a t r i x A we o b t a i n e d b e l o n g s tOx*4p. Now we c a n p r o c e e d t o p r o v e Theorem 2.6. S u f f i c i e n c y f o l l o w s f r o m d e f i n i t i o n o f F o r n e c e s s i t y , we p r o v e by i n d u c t i o n on _ 24 -k k n. I f any o f t h e i n e q u a l i t i e s J P i x i i I p i y i ' k = l , . . . , n - l i = l i = l i s a c t u a l l y an e q u a l i t y , t h e n t h e h y p o t h e s i s o f p r o p e r t y (3) o f i^p i s t r u e f r o m t h e i n d u c t i o n a s s u m p t i o n and we a r e done. O t h e r w i s e t a k e xo as i n Lemma 2.8 w i t h iuin k 6 = m l n { I p ^ y j - x )} >0. k = l , . . . , n - l i = l Then x ^ x ^ y on D . By t h e p r e c e d i n g argument x =y_A' , A !£s>4p. A l s o , f r o m Lemma 2.8, x=x^A", A" e >4p • Hence x=yA where A=A'A" e j 4 p . D. F u n c t i o n s p r e s e r v i n g t h e p - m a j o r i z a t i o n s . The f o l l o w i n g c h a r a c t e r i z e s t h e f u n c t i o n s t h a t p r e s e r v e t h e o r d e r i n g s : F o r ea c h p, l e t F ( : ; p ) : D — R be s u c h t h a t F (x ;p )-<F (y ;p) whenever x ^ y on D . Then we e x t e n d F ( * ; p ) t o R by F ( x ; p ) = F ( x ;p ) f o r x e D . C o n s e q u e n t l y , we see t h a t b e c a u s e x y on D 1 T=^x T r ^ ' " / o n D = ^ - F ( x ; p ) = F C x 7 1 ; ^ ) < F ( y T T ;£ 1 T )=F(y ,p) , so t h a t F p r e s e r v e s t h e o r d e r i n g s i n d e p e n d e n t l y o f t h e p e r m u t a t i o n i n v o l v e d . I f we d e n o t e by ^p t h e c l a s s o f a l l s u c h f u n c t i o n s , & t h e c l a s s o f a l l i n c r e a s i n g f u n c t i o n s and $ the c l a s s o f a l l d e c r e a s i n g f u n c t i o n s , t h e n we have t h e f o l l o w i n g : Theorem 2.9- F p r e s e r v e s a l l (%.) i f f F e UpOj ( 9p*0i&0-The p r o o f i s s i m i l a r t o t h a t o f S c h u r convex f u n c t i o n . I n c a s e o f d i f f e r e n t i a b l e f u n c t i o n s , we c a n c h a r a c t e r i z e - 25 -f u n c t i o n s o f 3p by t h e i r d e r i v a t i v e s . Theorem 2.10. I f F i s d i f f e r e n t i a b l e , t h e n F e 5"p i f f F (2) ( x . - x . ) ( — - — > 0 f o r a l l i , j = l , . . . ,n s a t i s f i e d : (1) F ( x ; p ) = F C x ^ p 7 7 ) f o r e v e r y p e r m u t a t i o n T  , i " X J M P ± § x i " P j § x j P r o o f : F i r s t s u p pose F e ^ p , t h e n (1) f o l l o w s f r o m t h e d e f i n i t i o n . To p r o v e ( 2 ) , assume w i t h o u t l o s s o f g e n e r a l i t y t h a t x e D . F o r 6>0 and i < j , l e t ^ V p T + p T ' • • • ' x i + p T + p 7 ' x i + l + p T ' • ' ' X J - 1 + P 7 ' x J >••• ' x n } ' X ' = ( X I + P T + P 7 ' • • > x i - i + p T + p T > x i + p T > • • • > V I + P 7 ' X J •' • • >xn> • E ( x a i p ) - F ( x ' 3 p F ( x b ; p ) - F ( x ' ; p ) T a k i n g f — = l i m ^ • — , — = l i m _. .,_ — ; > 8 x i 6->0 + -V5/P, 3 X j 6 + 0 + 6/p i aw i aw F ( x a ; p ) - F ( x b ; p ) we have ^ i | _ _ 1_ = l i m , ~ ~ > 0 . • p 3 x p 3 x + fi Next suppose (1) and (2) h o l d . We o n l y need t o p r o v e t h a t F p r e s e r v e s ^ on D . B e c a u s e o f Theorem 2.5 , we may t a k e x = ( X l , x 2 ) , y^Cy^yp)? 2L Z o n D a n d 2L^ y • > t h e n we have y l > x l - x 2 > y 2 a n d P l x l + P 2 x 2 = p l y l + P 2 y 2 ' T h e r e f o r e we have P l ( ~ y l ~ X l ^ = p 2 ( y 2 - x 2 ) > 0 . F o r some e , 0.<6<1, - 26 -P ( x ; E ) - P ( I ; E ) = - {(y^ff-)^ (y2-x2)p-jj, where " j ^ " i n d i c a t e s e v a l u a t i o n at the p o i n t ( l - 9 ) x + 9 y Hence F ( x ; p ) - F ( y ;p_) = - P i ( y 1 - x 1 ) j f-J^  - ffj|Qj 5 °-C o r o l l a r y 2.11. I f f i s d i f f e r e n t i a b l e , then n r r F(x;p)= J p i f ( x 1 ) b e l ongs t o J p i f f f i s convex. J u s t l i k e Schur convex f u n c t i o n s , F e 3p r e s t r i c t e d t o t h e h y p e r p l a n e p^x^+...+p x =K a t t a i n s i t s minimum a t t h e p o i n t n n z=(K/ I p.,...,K/ 1 p.) where a l l components a r e e q u a l . I n 1=1 1 i = l 1 o r d e r t o see t h i s , i t i s s u f f i c i e n t t o show t h a t any p o i n t y £ D77 l y i n g on t h i s h y p e r p l a n e s a t i s f i e s z_ ^ y_ on D77. Wi t h o u t l o s s o f g e n e r a l i t y , we assume y e D. I f f o r some k, k = l , . . . , n - l , P-LY-L+. • -+P ky k < P l — — + . • - + P k ^ — , I p ? l I P i 1=1 1=1 t h e n P l y k + . . . + p k y k < p ^ t . . .+P ky k < P^-^— +• • - + P k — K I P ± I P i 1=1 1=1 n So we have y <...<y,<K/ £ p. . n _ _ k 1=1 1 T h e r e f o r e p.., y , . n + . . . + p v < p v + 1 + • • • +P K k + l ^ k + l * n ° n ^k+1 n *n n I P ± I P ± 1=1 1=1 - 27 -Summing the f i r s t and the l a s t i n e q u a l i t i e s , we a r r i v e a t the n c o n t r a d i c t i o n J P-Y- < K . Hence we con c l u d e t h a t z_ ^  y 1 = 1 1 1 on D . T h e r e f o r e , the f u n c t i o n P i n c r e a s e s from z_ a l o n g any r a y l y i n g on the h y p e r p l a n e P]_ x]_ + • • • + P n x n = ^ ' ^ e c-'- a s s 3 p i n c l u d e s the c l a s s o f Schur convex f u n c t i o n s c o r r e s p o n d i n g to the case p, = ... = p y l ^n As an i l l u s t r a t i o n , l e t us c o n s i d e r s t r a t i f i e d s a m p l i n g w i t h r s t r a t a , and w i t h N^,...N as the s i z e s of the s u b p o p u l a t i o n s . Suppose t h a t the v a r i a n c e w i t h i n each s t r a t u m 2 2 i s known ( s a y , e s t i m a t e d from p a s t e x p e r i e n c e ) as S-^,...VS . I f we draw samples n^,...,n from the c o r r e s p o n d i n g s t r a t a w i t h sample means y ^ , - - - , ^ , t h e n the e s t i m a t e o f the p o p u l a -t i o n mean i s y = ^ N.y./N , where N = J N. ; w i t h v a r i a n c e i = l 1 1 i = l r r V = (1/N 2) I N.(N.-n.)S?/n. = (1/N 2) £ N?S?/n. - c o n s t a n t . — 1=1 1=1 For f i x e d s a m p l i n g c o s t c ] _ n ] _ + • • • +Crnv = ^ ' ^ e ° b J e C T J i v e i s to choose s a m p l i n g u n i t s n so as to m i n i m i z e the, v a r i a n c e , 2 2 or e q u i v a l e n t l y the sum J N.S./n. . With p. = N . S ./c~7 , • - i 3_ 1 1 • — v ' 1 1 1 1 1=1 2„2 x. = n./cT/N.S. and f ( x ) = 1/x , we have N7S7/n. = p . f ( x . ) I 1 1 1 1 ' i i i ^ 1 1 r S i n c e f i s convex, t h e minimum of £ p . f ( x ± ) I s a t t a i n e d i = l - 28 -where x, = ... = x , i . e . n n/c"T/N, S,. = ... = n /c /N S_ 1 r 5 1 1 1 1 r r r r , a g r e e i n g w i t h the r e s u l t o b t a i n e d u s i n g the Lagrange m u l t i p l i e r t e c h n i q u e . One advantage o f u s i n g t h i s approach i s t h a t we have p a r t i a l o r d e r i n g s d e f i n e d on x such t h a t x ^ x' o n ^ V ( x ) < V(x') , where V(x) i s the v a r i a n c e c o r r e s p o n d i n g t o the s a m p l i n g u n i t s n a s s o c i a t e d w i t h x . F u r t h e r m o r e , because f i s d e c r e a s i n g , we can s t r e n g t h e n the above r e s u l t by x % x' on D^==>V(x) < V ( x ' ) . - 29 -Chapter 3 A. I n t r o d u c t i o n . I n t h i s c h a p t e r , we extend the m a j o r i z a t i o n s t o t h e i r s t o c h a s t i c v e r s i o n s and d i s c u s s the r e l a t i o n s among them. We d e a l w i t h t h e p r e s e r v a t i o n theorem o f Schur c o n v e x i t y by Pro s c h a n and Sethuraman (1977) and o f m o n o t o n i c i t y , o b t a i n i n g a c l a s s o f i n e q u a l i t i e s t h a t a r i s e i n some m u l t i v a r i a t e d i s t r i b u t i o n s . Then we a p p l y t h e p r e s e r v a t i o n o f Schur con-v e x i t y 'to c o n s t r u c t a c l a s s o f t e s t f u n c t i o n s i n one t y p e o f h y p o t h e s i s t e s t i n g . B. S t o c h a s t i c v e r s i o n s o f m a j o r i z a t i o n s . The f o l l o w i n g n o t i o n s o f s t o c h a s t i c o r d e r i n g between r e a l random v a r i a b l e s a re used repeatedly.. D e f i n i t i o n 3 . 1 - (1) X i s s t o c h a s t i c a l l y l e s s t h a n Y , de-noted by X stjc Y , i f f f o r every r e a l t , Pr {X>%}<Pr {Y>t }, i . e . P ( t ) < F _ _ ( t ) . (2) X = S i f f X st<Y and Y st< X. The f o l l o w i n g i s w e l l known. P r o p o s i t i o n 3 -2 . I f X st<Y, t h e n EX<EY., p r o v i d e d t h e expecta-t i o n s are f i n i t e . rO P r o o f : EX= -Pr{X<t}du ( t ) +• P r { X > t } d y ( t ) o •'0 - 30 -and EY = - P r { Y < t } d y ( t ) + ° P r { Y > t } d y ( t ) , 0 where y stands f o r t h e Lebesque measure or* the c o u n t i n g measure. The i n e q u a l i t y EX < EY now f o l l o w s from the i n e q u a l i t i e s o f the c o r r e s p o n d i n g i n t e g r a n d s . I n t he case of degenerate random v a r i b l e s , t he s t o c h a s t i c i n e q u a l i t y reduces t o o r d i n a r y i n e q u a l i t y . S i m i l a r l y , v a r i o u s s t o c h a s t i c e x t e n s i o n s of m a j o r i z a t i o n a r e p o s s i b l e w h i c h , f o r degenerate random v e c t o r s , reduce t o o r d i n a r y m a j o r i z a t i o n . These e x t e n s i o n s , proposed by v a r i o u s a u t h o r s , a r e o b t a i n e d from t h e f o u r e q u i v a l e n t c o n d i t i o n s o f Theorem 1.16 by chan g i n g the c o n s t a n t s t o random v a r i a b l e s and r e p l a c i n g t he i n e q u a l i t i e s e i t h e r by those o f e x p e c t a t i o n s , or by s t o c h a s t i c i n e q u a l i t i e s . However, t h e s e v e r s i o n s t u r n out t o be q u i t e d i f f e r e n t except under c e r t a i n imposed c o n d i t i o n s , some or a l l of them become e q u i v a l e n t . These v e r s i o n s a r e : X Y w i t h p r o b a b i l i t y 1 . F(X) < F(Y) w i t h p r o b a b i l i t y 1, f o r every Schur convex F. EF(X) < EF(Y) f o r every Schur convex F . F(X) st< F(Y) f o r every Schur convex F . EF(X) < EF(Y) f o r every symmetric convex F . : E {f ( X 1 ) + . . .+f (X )} < E { f ( Y 1 ) + . . . + f (Y )} f o r every convex f, M l M 2 M3 M4 V j / E i ] st<- j/t - i ] f o r k=1---n-1 a n d j / c i ] = j / c i ] where X r, n>...>X r -, i s the r e v e r s e d o r d e r s t a t i s t i c . [1]- - [n] - 31 -* n n n n M : I Y r i 1 t s t < I X f -, f o r k = l . . . . , n - l and £ Y m = J X r i=k L i=k 1=1 1=1 M 6: ( E X [ 1 ] 3 . . . , E X [ n ] ) ^ ( E Y [ i r . . . 3 E Y [ n ] ) . We can a l s o e x t e n d t h e two weak m a j o r i z a t I o n s *w* and ^ s i m i l a r l y w i t h s l i g h t m o d i f i c a t i o n . For st-< (st^O, we r e p l a c e •< w by ^ (-«( ) and f u n c t i o n s by i n c r e a s i n g ( d e c r e a s i n g ) f u n c t i o n s i n the above v e r s i o n s except f o r v e r s i o n 5- We s h a l l denote the r e s u l t i n g v e r s i o n s o b t a i n e d by P^,...,Pg (Q-^, • . . ,Qg) . For v e r s i o n 5, we have: k k r-: y Xr-.-, st< 7 Yr.-> f o r k = l , . . . , n . P n n V I Y r i i s t i I x r u f o r k = 1>---> n 5 i=k L 1 J i=k L 1 J The f o l l o w i n g two r e s u l t s c o n c e r n i n g t h e r e l a t i o n s among t h e s e v e r s i o n s a r e due t o M a r s h a l l and O l k i n ( t o a p p e a r ) . t ! P r o p o s i t i o n 3*3- and M^ are e q u i v a l e n t , so are M^ and M 2 i P r o o f : The e q u i v a l e n c e o f and M-^  f o l l o w s from t h a t o f x -< y and F ( x ) < F ( y ) f o r a l l Schur convex f u n c t i o n s . F . i M 2 ==^ - M 2 f o l l o w s from P r o p o s i t i o n 3«2. Now suppose M 2 h o l d s . F o r a r b i t r a r y t and Schur convex f u n c t i o n F , l e t S = { z_: F (z) >t} . Then l g i s Schur convex T h e r e f o r e P r { F ( X ) > t } = E l s ( X ) < E 1 (Y) = P r { F ( Y ) > t } . Hence M2=^-M, t - 32 -Theorem 3-4. M S M, No f u r t h e r i m p l i c a t i o n i s p o s s i b l e . 1 ^ 2,_: 5 For the p r o o f and. t h e co u n t e r e x a m p l e s , see M a r s h a l l and O l k i n (to a p p e a r ) . P r o p o s i t i o n 3 • 5 • I f X^,...,X n are i . i . d . , Y ^ . . . ^ a r e i . i . d . and X,+...+X^ = Y-. + ...+Y , th e n X. = Y. . I n p a r t i c u l a r , i f 1 n. 1 n' i i ^ ' X 1 , . . . J X n are i . i . d . , Y 1 3...,Y n a r e i . i . d . , t h e n M-^M^M^ and M,_ are e q u i v a l e n t . P r o o f : Denote the d i s t r i b u t i o n s o f X.,Y. by and $,r . l 5 l J X Y X 1 +...+X n =" Y 1+...+Y n=> [$x0n = [ $ y ] n = > $ x = $ y = > *X ~ ^ Y ' where $' i s the F o u r i e r t r a n s f o r m o f $ . The e q u i v a l e n c e o f IY^ ,M2 ,M,_ ,M,_ r e s u l t s from : M n=> M^^r^M,- or M* r=b X n + . . .+X = Y n + ...+Y . 1 r 2 7 5 5 r 1 n 1 n I t may appear t h a t M,- and M a r e c l o s e l y r e l a t e d , so t h e i r e q u i v a l e n c e r e q u i r i n g such s t r o n g c o n d i t i o n s i s a b i t s u r p r i s i n g , F o r s t o c h a s t i c weak m a j o r i z a t i o n s , s i m i l a r r e s u l t s can be o b t a i n e d . The p r o o f s a r e easy e x t e n s i o n s o f those of the t h r e e p r e v i o u s r e s u l t s . - 33 -P r o p o s i t i o n 3-6. a n d a r e e q u i v a - l e n t > s o a r e P 2 ( Q 2 ) and P 2 ( Q 2 ) . Theorem 3.7. p p P l = > P2 . \ No f u r t h e r i m p l i c a t i o n i s p o s s i b l e . The same i s t r u e i f we r e p l a c e the P's by t h e Q's. P r o p o s i t i o n 3.8. I f X^,...,X a r e i . i . d . , Y-^,...,Yn a r e i . i . d . , and X 1+...+X n st< v Y-j^t. . .+Yn, t h e n X 1 st< Y-L. I n p a r t i c u l a r , i f X-,,...,X are i . i . d . , Y-,,...,Y are i . i . d . , t h e n P n ,P n and Pr 1' ' n ' 1 5 ' n ' 1' 2 5 (Q-|_,Q2 and Q^) are e q u i v a l e n t . Because v e r s i o n 1 i s too s t r o n g t o be s a t i s f i e d i n most s i t u a t i o n s of i n t e r e s t , we s h a l l use v e r s i o n 2 as our d e f i n i t i o n f o r s t o c h a s t i c m a j o r i z a t i o n and weak m a j o r i z a t i o n s (as i n N e v i u s , P roschan and Sethuraman 1977)- Prom now on, v e r s i o n 2 w i l l be assumed i f t h e v e r s i o n i s not e x p l i c i t l y s p e c i f i e d . V e r s i o n 3 has the f o l l o w i n g i l l u s t r a t i o n : P r o p o s i t i o n 3-9- I f . X ^ , . . . , X a r e exchangeable random v a r i a b l e s and u -< v, t h e n (u,X, ,. . . ,u X ) st-< (v,X,,...,v X ) i n t h e sense — ^ —' 1 1 ' ' n n' 1 1 ' ' n n of v e r s i o n 3. The same i s t r u e i f we r e p l a c e «< by (\ ) and ( w, w* s t < by st-£ (st\ )• P r o o f : I t s u f f i c e s t o prove t h e case n=2 because of Theorem 1, I n t h i s c a s e , ( u ^ , u 2 ) "4. ( V ] _ J v 2 ^ — — ( u - ^ U g ) =a (v-^ , v 2 )+cT( v 2 , v-^) - 34 -f o r some a,a>0, a+a=l- For any symmetric convex f u n c t i o n F , E F ( u 1 X 1 , u 2 X 2 ) = E F [ ( a v 1 + a V 2 ) X 1 , ( a V 1 + a v 2 ) X 2 ] = E F [ a ( v 1 X 1 3 v 2 X 2 ) + a ( v 2 X 1 , v 1 X 2 ) ] < a E P ( v 1 X 1 , v 2 X 2 ) + a E P ( v 2 X 1 , v 1 X 2 ) = EF ( v ^ , v 2 X 2 ). The p r o o f f o r t h e P,Q v e r s i o n s a r e s i m i l a r , except w i t h t h e a d d i t i o n o f m o n o t o n i c i t y . The M^ v e r s i o n o f the above p r o p o s i t i o n i s proved by M a r s h a l l and P r o s c h a n (1965), who a l s o i l l u s t r a t e by an example t h a t f u r t h e r e x t e n s i o n t o M 2_,is not p o s s i b l e . The P^ v e r s i o n i s proved by Chong (1976). I n t u i t i v e l y , X st< j_Y-.me.ansr t h a t T Y';r.t.ends t o / m a j o r i z e X. . an i l l u s t r a t i o n , c o n s i d e r a b i n o m i a l random v a r i a b l e X w i t h parameters N, p where p > 1/2, t h e n X tends t o t a k e the I n t e g r a l v a l u e s i n [N/2,N] r a t h e r t h a n i n [0,N/2]; or e q u i v a -l e n t l y , X tends t o be g r e a t e r t h a n N-X. C o n s e q u e n t l y , (X+c,N-X) tends t o m a j o r i z e (X,N-X+c) f o r any c>0. I n f a c t , f o r any Schur convex f u n c t i o n F , N N EF(X+c ,N'--X) = I P(x+c ,N-x)Pr{X=x} = J F (N-x+c ,x )Pr{X=N-x} x=0 x=0 so EF(X+c,N-X) = i I [F(x+c,N-x)Pr{X=x}+F(N-x+c,x)Pr{X=N-x}] . (1) x = 0 S i m i l a r l y , EF(X,N-X+c) N = i I [F(x,N-x+c)Pr(X=x}+F(N-x,x+c)Pr{X=N-x}] (2) d x = 0 - 35 -< But F(x+c,N-x) = F(N-x,x+c) = F(N-x+c,x) = F(x,N-x+c) > x = N/2 4=i>Pr{X=x} = Pr{X=N-x} , < > so t h a t each summand i n (1) i s g r e a t e r t h a n t h e c o r r e s p o n d i n g §n<B i n (2). Hence (X,N-X-c )st-< (X+c,N-X), which a l s o means t h a t F(X,N-X+c) st< F(X+c,N-x) f o r any Schur convex f u n c t i o n F. I n p a r t i c u l a r , i f f i s convex, t h e n F ( x ^ , x 2 ) =f (x-^ ) + f (x^) i s Schur convex, and f(X)+f(N-X+c) st< f(X+c)+f(N-X) , which i s a lemma proved by Cohen and S a c k r o w i t z (1975)• C. P r e s e r v a t i o n o f Schur c o n v e x i t y . Now we come t o the p r e s e r v a t i o n theorem o f Schur f u n c t i o n s , F i r s t we i n t r o d u c e two d e f i n i t i o n s . D e f i n i t i o n 3.10. For A C R, I J J ( X , X ) : A X R — * - R i s s a i d t o be t o t a l l y p o s i t i v e o f o r d e r 2 ( T P 2 ) i f (1) i|>U,x) > 0, (2) f o r X-j^jXgeA, A 1 ^ X 2 and 0<x 1<x 2, ^ ( X 1 , x 1 ) i | j ( X 2 ,x 2) > i J ; ( X 1 , x 2 ) i ( ; ( X 2 , x 1 ) . D e f i n i t i o n 3-H- I f A C R i s such t h a t X-j^, X 2eA, X-j^>X2 X 1±X 2eA , t h e n i | ; ( X , x ) : A x R — R i s s a i d t o s a t i s f y t h e semi-group p r o p e r t y ( i ^ / i s SGP) i f (1) ^(X>,x)=0 f o r a l l xsO, (2) f o r any £ , X 2eA ,T\> ( X 1+X 2 ,x) = i|» (\ 1 ,x-y )ty ( X 2 ,y ) dy (y) , - 36 -where y i s Lebesque measure or c o u n t i n g measure, Prom now on, we s h a l l omit the dummy i n d e x i n the summation I and t h e pr o d u c t ~J~|" whenever I t i s c l e a r from t h e c o n t e x t . The f o l l o w i n g i s the p r e s e r v a t i o n theorem o f P r o s c h a n and Sethuraman (1977). Theorem 3-32. I f ip U ,x ) : AxR —> R i s T P 2 and SGP, B i s a non-n e g a t i v e f u n c t i o n o f two v a r i a b l e s , t h e n f o r any Schur convex f u n c t i o n F , H(A_) d e f i n e d by H(X) = F(x)B(£ X l JX±JrFr ( X 1 , x i ) d i i ( x . 1 ) . . .dy ( x n ) i s Schur convex i n A_ , where the i n t e g r a l i s assumed t o e x i s t . P r o o f : I t s u f f i c e s t o prove the case n=2 because of Theorem 1.8 I f (X£,X 2) < ( x j , X 2 ) , t h e n X - ^ X ^ x j + X ^ . t i Hence G ( x 1 3 x 2 ) = F(x^,x 2)B(x^+x 2 , X - ^ + X 2 ) = F (x ±, x 2 )B ( x 1 + x 2 , X-L+X 2 ) i s Schur convex i n ( x ^ , x 2 ) Because G i s symmetric, i t i s easy t o v e r i f y t h a t H i s a l s o t t symmetric. T h e r e f o r e we may assume t h a t X1>X-^>X2>X2 . Now H(X-j_, X 2 ) - H ( X 1 , X 2 ) I T G(x 1-,x 2) [ i p ( X 1 , x 1 ) i J ; ( X 2 , x 2 ) - i J ; ( X 1 , x 1 ) i J j ( X 2 , x 2 ) ]dy ( x 1 ) d y ( x 2 ) t G ( x 1 5 x 2 ) L>(X >^1-Y )ty ( X 2 , x 2 ) - i j j ( X 1 , x 1 ) i ^ ( X 2 , x 2 - y ) ] t xijj ( X - ^ A ^ y ) d y ( x 1 ) d y ( x 2 ) d y (y) ty ( X 1 - X 1 , y ) [ i J ; ( X 1 , x . 1 ) ^ ( X 2 , x 2 ) - i r ( X 1 , x 2 ) i r ( X 2 . x 1 ) ] x [ G ( x 1 + y i x 2 ) - G ( x 1 , x 2 + y ) ] d y ( x ± ) d y ( x 2 ) x-^>x2 - 37 -> 0 •> where t h e second e q u a l i t y f o l l o w s from b e i n g SGP and \^-\^=\2' the t h i r d e q u a l i t y f o l l o w s from c a n c e l l i n g t he p a r t x-^=x^, c h a n g i n g v a r i a b l e s and G b e i n g symmetric; t h e l a s t i n e q u a l i t y f o l l o w s from b e i n g T P 2 J G Schur convex and x ^ > X 2 , r e s u l t i n g i n the i n t e g r a n d b e i n g n o n - n e g a t i v e . Because F i s Schur convex i f f -F i s Schur concave, Theorem 3.32 i s a l s o t r u e i f we r e p l a c e Schur c o n v e x i t y by Schur c o n c a v i t y . F u r t h e r m o r e , from Mudholkar's r e s u l t (1966), F(x-^,x^)=f (x-^)f (x^) i s Schur convex (concave) I f f f i s l o g -convex ( l o g - c o n c a v e ) . So we have the f o l l o w i n g c o r o l l a r y : C o r o l l a r y 3 . 1 3 - ( P r o s c h a n and Sethuraman 1974). If-tifi (.A ,,x) :.is TP^ and SGP and f i s log - c o n v e x ( l o g - c o n c a v e ) , t h e n h(X) d e f i n e d by h(X). = \p ( X ,x )g (x )dy (x) i s lo g - c o n v e x ( l o g - c o n c a v e ) . We g i v e below some commonly encountered f u n c t i o n s t h a t a r e w e l l known t o be TP^ and SGP. V e r i f i c a t i o n w i l l be o m i t t e d . P r o p o s i t i o n 3-14. The f o l l o w i n g ty(X,x) a r e TP^ and SGP: Xx (1) 4,Q,x) = A_ f o r x = 0 , l , 2 , . . . , A>0, A. • (2) i>(X,x) = f o r x = 0 , l , 2 , . . . , A=l,2,..., here we adopt the c o n v e n t i o n t h a t ( y ) = 0 i f x>X, vlxj X'' (3) T|>U,X) = ^ y - y f o r X>0 , A>0, - 38 -(4) ip(\,x) = Tx\r\l) f o r x = 0,.l,2,... , X>0, w i t h "ijj(x,x)=0 o t h e r w i s e " b e i n g assumed. I n case t h a t a) (x;X. ) =cB(,j^X^ )7T^ (X^ jX^ ) i s the d e n s i t y o f a random v e c t o r X 5 H(X.) i s the e x p e c t a t i o n E ( F ( X ) ) . Theorem 3-12 A X_ t can be w r i t t e n i n the form X^  X_ = ^ X st-< X , . F o r c o n v e n i e n -X X_ ce, we i n t r o d u c e the f o l l o w i n g : D e f i n i t i o n 3-15. A f a m i l y o f n - d i m e n s i o n a l d i s t r i b u t i o n f u n c t i o n s {$,:XeA n} ( o r random v e c t o r s { X A : X e A n } ) i s s a i d t o be A_A A_ a Schur convex f a m i l y ( i n X) i f H(X)=E ( F ( X ) ) i s Schur convex i n A _X whenever F i s Schur convex. P r o p o s i t i o n 3-1^ ". {$,} i s a Schur convex f a m i l y i f X =(X , A A. ]]_ X ) i s composed of independent components o f t h e same f a m i l y , n and X 1=1,...,n has d e n s i t y from e i t h e r o f the f o l l o w i n g A i f a m i l i e s : (a) P o i s s o n ( R i n o t t 1973): XX -X <S(x;X) = f o r x = 0,l,2,..., X>0 . (b) B i n o m i a l ( N e v i u s , P r o s c h a n and Sethuraman 1977): $(x;A) = ( A ) 0 X ( 1 - 8 ) A _ X f o r x=0,l,2,..., X=l,2,..., and f i x e d 9>0. (c) Gamma ( N e v i u s , P r o s c h a n and Sethuraman 1977): - 39 -cj>.(x,A) = QTX(X) E 6 X F O R X > ° 3 A>0 and f i x e d e>0. Pr o o f : T h i s i s because the j o i n t d e n s i t y jy<|>(x A ^ ) i s cB ( J x ^ , £ A ^ )"TT^ ( > x i ^ w i t h ip i n the form shown i n P r o p o s i t i o n 3-14 P r o p o s i t i o n 3-17. {*,} i s a Schur convex f a m i l y I f X has d e n s i t y A A^  i n t he f o l l o w i n g form: (1) M u l t i n o m i a l ( R i n o t t 1973): , x. <|>(x;X) = N I T J - ^ x. l f o r x ± = 0 , l , . . . ,N 3 £ x ± = N , A 1>0, l\±=l. (2) D i r i c h l e t ( N e v i u s , P r o s c h a n and Sethuraman 1977): f o r x i>0, J x 1 < l , A 1>0, and f i x e d 3>0. (3) I n v e r t e d D i r i c h l e t ( H o l l a n d e r , P r o s c h a n and Sethuraman, t o a p p e a r ) : r (e+£A, ) x , x i _ 1 r ( e ) ( i + I x . ) e + ^ i " T r ^ y f o r x 1>0, A 1>0, and f i x e d 9>0. (4) N e g a t i v e m u l t i n o m i a l ( M a r s h a l l and O l l t i n , t o appear) x. r(k+Ix ) A . A i <t>(x;A) = —d-I^TT^— f ( k ) 1 x,! - 40 -f o r x 1 = 0,l,2, . . . , Xj_>0, J x 1<l.and f i x e d k>0. M u l t i v a r i a t e n e g a t i v e b i n o m i a l ( N e v i u s , P r o s c h a n and Sethuraman 1977) : r(N) x ± ! f o r x 1 = 0,l,2. . . . , A±>0, and f i x e d N>0. D i r i c h l e t compound n e g a t i v e m u l t i n o m i a l ( H o l l a n d e r , P r o s c h a n and Sethuraman, t o a p p e a r ) : r(N+J x,)r(e+yx,)r(N+e) r ( x . + A , ) <Kx;X) = i i TT r(N ) r ( e ) r(N + e + ^ A 1+^x i) x 1 : r ( x 1 ) f o r x i = 0,l,2, . . . , A±>0; and f i x e d N>0, @.>0. M u l t i v a r i a t e h y p e r g e o m e t r i c ( N e v i u s , P r o s c h a n and Sethuraman 1977): i f o r x i = 0 , l , . . . , },'x1=N, A i=l,2,..., J A ^ M ; and f i x e d p o s i t i v e i n t e g e r s M, N w i t h M>N. M u l t i v a r i a t e i n v e r s e h y p e r g e o m e t r i c : -1 <f>(x,A) = M V / M - j A .\ 1 4 M-£ A .-k+1 #X . 1" Mx§ l x . / ^k+yx.-l*7 * k - l 1 M-yx.-k+l' ' Vx J ^ 1 L 1 '1 f o r x.=0,1,2,...,X =1,2,... , JX^M-k; and f i x e d p o s i t i v e i n t e g e r s M,k w i t h k<M. Ne g a t i v e m u l t i v a r i a t e h y p e r g e o m e t r i c ( N e v i u s , P r o s c h a n and Sethuraman 1977): - 41 -• N.!.r(M) r(x.,+x,) d,'(x3A) = , TT——— r(N+M) x . ! r ( A ) ) 1 1 f o r x 1=.0,l,2, . . . , Jx±=N, A i>0, ^A±=M; and f i x e d M>0, and p o s i t i v e i n t e g e r N . The p r o o f f o l l o w s d i r e c t l y from Theorem 3-32 and P r o p o s i t i o n 3'. 14 The d i s t r i b u t i o n s (1), (7) and (9) a r e ( n - 1 ) - d i m e n s i o n a l because x i s c o n f i n e d on t h e h y p e r p l a n e x^+...+xn=N. P r o p o s i t i o n 3.18. (1) I f {$,} i s a Schur convex f a m i l y , A t h e n Pr{r- L< (or< )X^< (or< ) r 2 f o r a l l i|A} i s a Schur concave f u n c t i o n of A_, where r-^>-oo} r2|<». (2) I n a d d i t i o n , i f X^ i s non--negative and Z i s t h e number o f A A_ z e r o components of X.. , t h e n A -< A' =^ Z st<Z , ~~A A — A. Pr o o f : (1) I t i s easy t o v e r i f y t h a t F ( x ) d e f i n e d by (1 i f min{x . }> (or> ) r 3 and max.{x. }< (or< ) r 0 P(x) =\ i 1 " 1 1 _ 1 0 o t h e r w i s e i s Schur concave. T h e r e f o r e E^F(X) = P r { r 1 < ( o r < ) X i < ( o r < ) r 2 f o r a l l i|A} i s Schur concave i n \. (2) T h i s f o l l o w s from c o n s i d e r i n g t h e i n d i c a t o r f u n c t i o n s of {x>0:. 1% = 0 ) < k } H> 0, i w h ich a r e Schur convex on - 42 -The m u l t i n o m i a l case of (1) w i t h r2=°° co r r e s p o n d s t o t h e r e s u l t o f O l k i n (1972) and w i t h r1=-<» co r r e s p o n d s t o t h a t o f R i n o t t (1973)- The m u l t i n o m i a l case of (2) c o r r e s p o n d s t o the r e s u l t o f Wong and Yue (1973). D. P r e s e r v a t i o n of m o n o t o n i c i t y . For t h e e x t e n s i o n t o s t o e h a s t i c i w e a f c ^ m a j o r i z a t i o n , we want X ^ ( X ) X' l x strfr ( s t ^ ) Xx , < > E A F ( X ) < E X , F ( X ) f o r every Schur convex i n c r e a s i n g ( d e c r e a s i n g ) F. Thus, i n a d d i t i o n t o the p r e s e r v a t i o n of Schur c o n v e x i t y , we need a p r e s e r v a t i o n o f m o n o t o n i c i t y as w e l l . Theorem 3.£9» L e t cfj (x ;X_) be a d e n s i t y of the form cj)(x,X) = c B ( [ x i , J X 1 ) T T ^ ( X i , x . ) , where y i s SGP and B s a t i s f i e s : B(x,X) = B(x+y,X+a)^(a,y)dy(y) , whenever a,XeA, a>0. Then t h e t r a n s f o r m a t i o n F — H d e f i n e d by H i A ) = ••• F ( x ) * U J I ) dU (*-,_). . . d y ( x n ) J J p r e s e r v e s m o n o t o n i c i t y , i . e . i f F i s i n c r e a s i n g or d e c r e a s i n g , t h e n so i s H . Pr o o f : Suppose F i s i n c r e a s i n g . We o n l y need t o show t h a t H i s i n c r e a s i n g i n i t s k^ b component f o r k = l , . . . , n . For a,X ieA and a>0, - 43 -H(X-|_> • • • j A k + a , . . . ,X n) = c = c >c F ( x ) B ( ^ x ± j A 1 + a ) m ( X ± 3 x 1 ) 1 j J ( A k + a 3 x k ) J T d y ( x 1 ) F(x)B(£x. J X . + a ) T T ^ ^ 1 3 x 1 ) ^ ( x k J x k - y ) ^ ( a 3 y ) T T d V i ^ x i ) d y ^ ) i ^ k F(x 1, . . . , x k + y , . . . ,x n)B( Jx.±+y ,^X1+a)TT^(X,x1)4)(a,y) x^pdy (x1)dy(y) F ( x ) B ( j x ± + y j x i + a ) T T ^ ^ i ' x i H ( a 5 y ) T T d y ( x i ) d v i ( y ) F ( x ) B ( J x . , J x 1 ) T T ^ ( A 1 3 x . ) T T d y ( x . ) = H(X) , where t h e s e c o n d e q u a l i t y f o l l o w s f r o m t h e SGP; t h e t h i r d e q u a l i t y f o l l o w s f r o m a c h a n g e o f v a r i a b l e ; t h e n e x t i n e q u a l i t y f o l l o w s f r o m F b e i n g i n c r e a s i n g a n d y>0 ( i n t e g r a n d i s 0 f o r y<0); and t h e n e x t e q u a l i t y f o l l o w s f r o m t h e h y p o t h e s i s on B . The c o n d i t i o n t o be s a t i s f i e d by B i s s i m p l y t h a t the m a r g i n a l d e n s i t y i s i n a s i m i l a r f a m i l y , o n l y w i t h a lower d i m e n s i o n ; n n n i . e . i f (|>x1 x (x, ,...,x ) = cB( £ x , Y \ ) TT^(X.,x ) , XT...Xn 1 n 1=1 1 1=1 1 1=1 1 n n - l n - l n - l t h e n <f>x-,.,,x (x ,. . . ,x ) = cB( [ x [ L ) J W x ^ x ) . X'1...Xn~,  1 n 1 1 = 1 1 1=1 1 1 = 1 1 n - l I n p a r t i c u l a r , i f the m u l t i v a r i a t e d e n s i t y i s a p r o d u c t o f u n i v a r i a t e d e n s i t i e s , each b e i n g SGP, t h e n t h e m u l t i v a r i a t e d e n s i t y i s of the form ( X^ , x^ ) , so i t p r e s e r v e s m o n o t o n i c i t y , F u r t h e r m o r e , i f <j>(x;X) s a t i s f i e s the h y p o t h e s i s of t h e above - 44 -theorem, so w i l l I t s m a r g i n a l d e n s i t i e s . P r o p o s i t i o n 3.20. The Schur convex f a m i l i e s of independent P o i s s o n , B i n o m i a l and Gamma o f P r o p o s i t i o n 3-26 p r e s e r v e m o n o t o n i c i t y . The p r o o f f o l l o w s from the f o l l o w i n g obvious lemma: Lemma 3.21. I f ip(X,x) i s T P 2 ( o r SGP), t h e n so are a ty(X,x) and b^i|j(A,x) where a,b>0. Now we g i v e some examples o f m u l t i v a r i a t e d i s t r i b u t i o n s t h a t s a t i s f y t he h y p o t h e s i s o f Theorem 3.19- Some appear s l i g h t -l y d i f f e r e n t from t h e i r p r e v i o u s forms shown i n P r o p o s i t i o n 3.17 because we r e q u i r e t he d i s t r i b u t i o n s t o be n - d i m e n s i o n a l . (1) M u l t i n o m i a l w i t h f i x e d N: (N-Vx.)! x.! f o r x i = 0 , l , 2 , . . . , Ix ±<N, ^ 1>0,IA 1<1. (1-X) N-x X x Here B(x,X) = T | ) ( X , X ) ( N - x ) ! (2) M u l t i v a r i a t e h y p e r g e o m e t r i c w i t h f i x e d M,N: f o r . x x = 0 s I y 2 ^ . . ; .^x^.S; XJ,=I,2/... . , £A ±<M . - 45 -Here B(x , x ) = . (JJ-x) ' ^ ( A ' x ) = ( x) ' Ne g a t i v e m u l t i v a r i a t e h y p e r g e o m e t r i c w i t h f i x e d . M,N: N I T ( M ) r(M+N-yx.-yx.) r(x . +x . ) < f , ( x ; A ) = ~ ~ TT r ( N + M ) ( N - J x 1 ) i r ( M - x 1 ) x i ! r ( x 1 ) f o r x 1 = 0 , l , 2 , . . . , Ix1<N, X ± > 0 , JA 1<M. T ( M + N - X - X ) r(x+x) Here B ( x , X ) = , ^ ( X , x ) = ( N - x ) ! T ( M - X ) x * r ( X ) M u l t i v a r i a t e n e g a t i v e b i n o m i a l w i t h f i x e d N : r ( N+Yx ) - N - J x X X i « > ( x j A ) = —d + I x . ) X J J — — r ( N ) 1 x ± : f o r x,=1,2. .. . ,AA.SO . x ' 3. x Here B ( x , X ) = r ( N+x) (1+ X ) i N ~ x , i |>(X ,x ) = — x! I n v e r t e d D i r i c h l e t w i t h f i x e d 6 : r(e+yx.) x / 1 - 1 d > ( x ; X ) = ^ TT— r(9)(i+£x.) 6 +2>i r ( x i } f o r i X(.:^0, X0>0 . 1 -L 1 Here B ( x , X ) = T< 9 + " X ) (1+x ) ~ 6 - X , . I | J ( X , X ) = * r(x) D i r i c h l e t compound n e g a t i v e m u l t i n o m i a l w i t h f i x e d N , r(N+9) rO+Jx. ) r ( N+Jx.) r(x . + x . ) j t -> \ v I N T' " 1 " 1 -t—r ± _L <j>(x;X) = J | r ( N ) r ( e ) r(N+9+JX 1+JX ±) x 1 : r ( x 1 ) f o r x.=0,l,2, .. . , 1X0>0 . l l - 46 -r (K+x )r (6+A ) r (X + A ) Here B ( X , A ) = , ty (X ,x) = r(N+e+x+x) x ! r ( A ) P r o p o s i t i o n 3.22. I f X has i t s d i s t r i b u t i o n i n the form o f (1) A to (6) d i s c u s s e d above, t h e n f o r m<n: 3 I ! (1) U 1 5 . . . , A m ) 4 ( ^ ) ( A 1 , . . . , A m ) (X-^, . . . ,X m| A-|_, . • . ,Xm) st>^ ( s t ^ ) (X-j^, . . . ,X m| A-j^, . . • ,A m) (2) Pr{p'1< (or< )X^< (or< ) r 2 f o r i = l, . . . ,m| A-^ , . . . ,A } i s a d e c r e a s i n g Schur concave f u n c t i o n o f (A-^,...,A m). (3) Pr {number o f zer o components o f (X^, . . . ,X m)>k| A-^ , . . . ,A m} i s a d e c r e a s i n g Schur convex f u n c t i o n o f (A-^,...,A m). P r o o f : (1) f o l l o w s from Theorem 3.13, 3.16 and t h e remark con-c e r n i n g t h e f u n c t i o n B . The p r o o f s f o r (2) and (3) f o l l o w t h e same l i n e as t h a t o f P r o p o s i t i o n 3-l8, t o g e t h e r w i t h the a p p l i -c a t i o n of (1). E. A p p l i c a t i o n t o h y p o t h e s i s t e s t i n g . C o n s i d e r a Schur convex f a m i l y , p a r a m e t r i z e d by A_, A^>0. Gi v e n A-L+...+An=nk where k i s e i t h e r a p o s i t i v e i n t e g e r or j u s t a p o s i t i v e number, depending on t h e form o f A, we want t o t e s t t he n u l l h y p o t h e s i s H Q: A1=...=A =k v e r s u s t h e a l t e r n a t i v e H-^ : not a l l the A^ e q u a l k' . I n t u i t i v e l y , symmetry o f the d i s t r i b u t i o n f u n c t i o n i m p l i e s t h a t i f the A^ a r e e q u a l , t h e n i t i s more p r o b a b l e t h a t the components X^ do not d i f f e r g r e a t l y , which a l s o means t h a t - 117 -X. / £x. do not d i f f e r g r e a t l y . F o r m u l a t e d I n terms o f Schur f u n c t i o n s , t h i s means t h a t when P i s Schur convex (Schur c o n c a v e ) , F ( X ^ / £ x . ,. . . ,X n/JXj. ) tends t o be s m a l l e s t ( l a r g e s t ) under H Q . So we can t a k e as a t e s t f u n c t i o n 0 < (>) where F i s some Schur convex (Schur concave) f u n c t i o n , 0<y<l and c a r e chosen so as t o a c h i e v e the d e s i r e d s i z e . I t i s easy t o v e r i f y t h a t T i s a Schur convex f u n c t i o n i n bo t h c a s e s , so t h a t the power f u n c t i o n 3T(A_) = E^T(X) i s Schur convex i n A. T h e r e f o r e , 3m a t t a i n s i t s minimum a t the n u l l p o i n t ( k , . . . , k ) . ' C o n s e q u e n t l y , the t e s t i s u n b i a s e d . The case o f the m u l t i n o m i a l f a m i l y , w i t h F ( x 1 / ^ X j , . . . » x n / £ x j ) - P(x 1/N,...,x n/N) = [ h ( x 1 ) , h convex, i s f i r s t found by Cohen and S a c k r o w i t z (1975). A g e n e r a l i z a -t i o n o f £h(x^) t o Schur convex f u n c t i o n s G(x) i s found by Perlman and R i n o t t ( t o a p p e a r ) . Cohen and S a c k r o w i t z (1975) a l s o p o i n t out t h a t t h e c l a s s J h ( x ^ ) i n c l u d e s t h e l i k e l i h o o d r a t i o t e s t w i t h h ( x ) = x l l o g x and the goodness o f f i t t e s t w i t h 2 h(x)=x . We can v e r i f y t h a t the goodness o f f i t t e s t w i t h 2 h{x)-x a l s o a p p l i e s t o the h y p e r g e o m e t r i c d i s t r i b u t i o n . - 48 -B i b l i o g r a p h y [1] Chong, K.M. (1976). An i n d u c t i o n theorem f o r r e a r r a n g e -ments. Canadian J o u r n a l of Mathematics 28, 154-160. [2] Cohen, A. and S a c k r o w i t z , H.B. (1975). Unbiasedness o f the C h i - s q u a r e , l i k e l i h o o d r a t i o , and o t h e r good-ness o f f i t t e s t s f o r the e q u a l c e l l case'. Annals of S t a t i s t i c s 3, 959-964. [3] Fuchs , L. (1947). A new p r o o f o f an i n e q u a l i t y of H a r d y - L i t t l e w o o d - P o l y a . • Matematisk T i d s s k r i f t B, 53-54. [4] Hardy, G.H., L i t t l e w o o d , J.E., and P o l y a , G. (1929). Some s i m p l e i n e q u a l i t i e s s a t i s f i e d by convex f u n c t i o n s . Messenger of Mathematics .,58 ,- -145-152. [5] Hardy, G.H., L i t t l e w o o d , J.E. and P o l y a , G. (1952). I n e q u a l i t i e s (second e d i t i o n ) . Cambridge U n i v e r s i t y P r e s s . [6] H o l l a n d e r , M. , P r o s c h a n , F.. and Sethuraman, J . ( t o ap--p e a r ) . F u n c t i o n s d e c r e a s i n g i n t r a n s p o s i t i o n and t h e i r a p p l i c a t i o n s i n r a n k i n g problems. [7] M a r s h a l l , A.W. and O l k i n , I . (to a p p e a r ) . I n e q u a l i t i e s v i a m a j o r i z a t i o n . [8] M a r s h a l l , A.W. and P r o s c h a n , F. (1965). An i n e q u a l i t y f o r convex f u n c t i o n s i n v o l v i n g m a j o r i z a t i o n . J o u r n a l of M a t h e m a t i c a l A n a l y s i s and A p p l i c a t i o n s 12, 87-90. [93 M i r s k y , L. (1959). I n e q u a l i t i e s f o r c e r t a i n c l a s s o f convex f u n c t i o n s . P r o c e e d i n g s of E d i n b u r g h Mathe-m a t i c a l S o c i e t y 11, 231-235. [10] M i t r o n o v i c , D.S. (1970). A n a l y t i c i n e q u a l i t i e s . S p r i n g e r - V e r l a g , New York. [11] Mudholkar, G.S. (1966). The i n t e g r a l o f an i n v a r i a n t unimodal f u n c t i o n over an i n v a r i a n t s e t - an i n e q u a l i t y and a p p l i c a t i o n s . P r o c e e d i n g s of American M a t h e m a t i c a l S o c i e t y 17, 1327-1333-- 49 -[12] N e v i u s , S.E., P r o s c h a n , P. and Sethuraman, J . (1977) Schur f u n c t i o n s i n s t a t i s t i c s I I : S t o c h a s t i c m a j o r i z a t i o n . Annals of S t a t i s t i c s 5, 263-273. A l s o , F l o r i d a S t a t e U n i v e r s i t y T e c h n i c a l Report M306, (1974). [13] N e v i u s , S.E., P r o s c h a n , F. and Sethuraman, J . (1974) Schur f u n c t i o n s i n S t a t i s t i c s I I I : A s t o c h a s t i c v e r s i o n o f weak m a j o r i z a t i o n , w i t h a p p l i c a t i o n s . F l o r i d a S t a t e U n i v e r s i t y T e c h n i c a l Report M307-[14] O l k i n , I . (1974). M o n o t o n l c i t y p r o p e r t i e s of D i r i c h l e t I n t e g r a l s w i t h a p p l i c a t i o n s t o the m u l t i n o m i a l d i s t r i b u t i o n and the a n a l y s i s of v a r i a n c e . B B i o m e t r i k a 59., 303-307-[15] O s t r o w s k i , A. (1952). Sur quelques a p p l i c a t i o n s des f u n c t i o n s convexes e t concaves au sens de I . Schur. J o u r n a l de Mathematiques Pure et a p p l i q u e e s 3_1, 253-292. [16] P e r l m a n , M.D. and R i n o t t , Y. ( t o appear) On t h e un-b i a s e d n e s s o f g o o d n e s s - o f - f i t t e s t s . [17] P r o s c h a n , F. and Sethuraman, J . (1977)- Schur f u n c t i o n s i n s t a t i s t i c s I : The p r e s e r v a t i o n theorem. F l o r i d a S t a t e U n i v e r s i t y T e c h n i c a l Report M254-RR. [18] R i n o t t , Y. (1973)- M u l t i v a r i a t e m a j o r i z a t i o n and rearrangement i n e q u a l i t i e s w i t h some a p p l i c a t i o n s t o p r o b a b i l i t y and s t a t i s t i c s . I s r a e l J o u r n a l o f Mathematics 15, 60-77. [19] Tomic, M. (1949). Gauss' theorem on the c e n t r o i d and i t s a p p l i c a t i o n . B u l l e t i n de l a S o c i e t e des M a t h e m a t i c i e n s et P h y s i c i e n s de l a R.P. de S e r b i e 1, 31-40. [20] Wong, C.K. and Yue, P.C. (1973). A m a j o r i z a t i o n theorem f o r the number o f d i s t i n c t outcomes i n N i n d e p e n -dent t r i a l s . D i s c r e t e Mathematics 6_, 391-398. 

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