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

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MAJORIZATION:  ITS EXTENSIONS  THE PRESERVATION  AND  THEOREMS  by  KOON WING CHENG B.So., U n i v e r s i t y  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 t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1977 ©  Koon Wing Cheng, 1977  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  agree  fulfilment  of  the  requirements  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make it I  in p a r t i a l  freely  available  for  his  t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f  of  this  representatives. thesis for  It  financial  this  thesis  The  gain s h a l l not  Mathematics  U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date  W t S L \  or  i s understood that copying or p u b l i c a t i o n  written permission.  Department o f  that  reference and study.  f o r s c h o l a r l y purposes may be granted by the Head of my Department by  for  be allowed without my  - i i-  Abstract.  T h i s t h e s i s deals mainly with the orderings majorizatIon, inequalities.  the two weak m a j o r i z a t i o n s  and t h e i r  One of these weak m a j o r i z a t i o n s  some a t t e n t i o n i n the l i t e r a t u r e .  these orderings  associated  has r e c e i v e d  However, the other one, being  dual to the former, i s t o t a l l y overlooked. which preserve  induced by  Schur f u n c t i o n s  are shown t o have a p p l i c a t i o n s i n  statistics. In Chapter 1, we d i s c u s s b r i e f l y t h e i r r e l a t i o n s with  the m a j o r i z a t i o n s and  Schur f u n c t i o n s , b r i n g i n g out the exten-  sions to the' weak m a j o r i z a t i o n s  where p o s s i b l e .  we g e n e r a l i z e the ordinary m a j o r i z a t i o n s by a v e c t o r p of p o s i t i v e components. t i e s o f these new m a j o r i z a t i o n s of o r d i n a r y m a j o r i z a t i o n s . generalized.  to those  parametrized  We d i s c u s s the proper-  i n a d i r e c t i o n p a r a l l e l t o that  The Schur f u n c t i o n s are l i k e w i s e  In Chapter 3, we d i s c u s s the s t o c h a s t i c  of m a j o r i z a t i o n s  extensions  and the p r e s e r v a t i o n theorem o f Schur con-  v e x i t y due to Proschan and Sethuraman (1977). p r e s e r v a t i o n theorem on monotonicity, extensions  In Chapter 2,  of the weak m a j o r i z a t i o n s .  With a  we study the s t o c h a s t i c Some i n e q u a l i t i e s  a r i s i n g 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 are found to be direct  consequences of the two p r e s e r v a t i o n theorems.  we c o n s i d e r  Finally,  the unbiasedness o f a c e r t a i n c l a s s of t e s t s of  significance.  - iii  Table of  -  Contents Page  Abstract Table of  ii Contents  iii  L i s t of F i g u r e  iv  Acknowledgement  V  Chapter 0 :  N o t a t i o n and c o n v e n t i o n s .  Chapter 1 :  A.  Introduction.  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  8  A.  Introduction.  B.  The 'p4jTiaj"drjr:za.fel;ona.  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.  F u n c t i o n s p r e s e r v i n g the  24  A.  Introduction.  B.  S t o c h a s t i c v e r s i o n s of  C.  P r e s e r v a t i o n of Schur c o n v e x i t y .  35  D.  P r e s e r v a t i o n of  42  E.  A p p l i c a t i o n to h y p o t h e s i s  Chapter 2 :  Chapter 3 :  Bibliography  .' 1  functions.  14 1  iat?  •&.  p-majorizations.  14  29  majorizations.  monotonicity. testing.  29  46 48  -  IV  -  L i s t of F i g u r e 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 for  h i s c o n s t a n t h e l p and  to the  supervisor,  encouragement.  Dr.  Finally,  I w i s h t o e x p r e s s my  E v e l y n Yang f o r h e r  t y p i n g of t h i s  A.W.  A l s o , I am  U n i v e r s i t y of B r i t i s h Columbia f o r the  assistance. Mrs.  t o t h a n k my  Marshall grateful  financial  gratitude  thesis.  to  - 1 -  0  Chapter  Notation For and  easier reading  conventions  be s p e c i f i e d  (1)  Real  the following notation Those  letters,  are denoted  by s m a l l  u s u a l l y by X,Y; v e c t o r s  multivariate  usually  a r e d e n o t e d by with  b y <j>  permutation  by f , g , hand  are u s u a l l y denoted  b y F,G,H.  Densi-  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 orj „  The d e c r e a s i n g  order  letters,  are underlined  are u s u a l l y denoted  functions  f o rboth  denoted  below  i n d i c a t e d by s u b s c r i p t s .  Univariate functions  ties  not l i s t e d  within the context.  u , v , x , y , z ; random v a r i a b l e s and v e c t o r s  components  (3)  thesis,  throughout.  numbers and v e c t o r s  capital  (2)  of this  a r e employed  will  by  and Conventions  • ]|>A,)^ifparametrised  order  arrangement  o f t h e components  statistics  LU  -  > ... >  x^,-. . . . x  X . -, > ... > X r  XJ-^-J  r  b y A_. x  |- j -is a n  o f x.  Reverse  -, a r e d e f i n e d  i n like  - LnJ  manner. (4)  I f TT i s a p e r m u t a t i o n x  the vector  (x  ,...,x  V (5)  Ordinary  (6)  Monotonicity  ordering  functions, in  (5).  of {l,...,n} ).  % of vectors  r e f e r s t o both  the latter  with  and matrices  i s componentwise.  u n i v a r i a t e and m u l t i v a r i a t e respect  A l s o , by i n c r e a s i n g  non-decreasing  , we d e n o t e b y  to the ordering  (decreasing)  (n'dn-increasing) .  we  shall  defined mean  - 2 -  (7)  The a b b r e v i a t i o n s  " i f f " f o r " i f and o n l y 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 " a r e 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 otherwise  stated.  -«  to  °°  unless  - 3 -  Chapter A.  Introduction. In t h i s  chapter, a brief  t i o n and Schur properties The  convex  survey of the concept  functions  of majorization w i l l  main theorem  majorizations.  introduced  i n Chapter  M a j o r i z a t i o n a n d weak  B u t some  be g e n e r a l i z e d  i n Chapter 2 .  will  equivalent  both l y i n g  conditions of  be e x t e n d e d  to their  3 •  majorizations.  Majorization i s a p a r t i a l ordering vectors  of majoriza-  i s discussed.  here gives  These c o n d i t i o n s  stochastic versions  B.  1  defined  f o r pairs of  on t h e same h y p e r p l a n e { x : x-^+:.:., •+x  n  = K }  in R . Definition 21 •< y >  1.1.  x i s said  t o be m a j o r i z e d by  y , i n notation  iff k  k [i]  [i]  for  k=l,...,n-l (1.1)  and l=l  [1]  1=1  Ei]  N o t e t h a t we h a v e t h e f o l l o w i n g  relations: n  (1)  the l a s t  e q u a l i t y i n (1.1) i s t h e same a s  n  j x. = J y. i=li=l  -  (2)  i f x •< Z  Z  a n c i  2L J t h e n  permutation of t h e i r (3)  4 -  i f x -< y J t h e n  x  y  d i f f e r by a  components;  x"^ y 71  f o r any p e r m u t a t i o n s  77  I f we e x t e n d t h e i n e q u a l i t i e s well,  and  TT  and  t o t h e l a s t e q u a l i t y as  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 .  Definition  1.2.  x i s said  t o be w e a k l y  k x ^ y , i f f J ril i=l  notation  x  We c a n r e w r i t e t h e s y s t e m n I  <  m a j o r i z e d by  k ^ ^fil i=l  ^  y ,in  . . . ,n .  o r  ( 1 . 1 ) i n a n o t h e r m a n n e r , as  n x  m  >  I yrn  i=k  i=k  n  n  f  o  r  k = 2  >--->  n  L  and T h i s s u g g e s t s . , 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  type: n  Definition  1. . 3  x % y i f fj x[Jj i=k  n > j i=k  E a c h o f t h e s e weak m a j o r i z a t i o n s , a p a r t i a l o r d e r i n g on R . n  by  f  o  r  k  =  1  n  ' and  '  , also defines  Unlike majorization, vectors  related  t h e s e o r d e r i n g s n e e d n o t be c o n f i n e d t o a h y p e r p l a n e . Trivial results  .  f o l l o w i n g from these d e f i n i t i o n s a r e :  - 5 -  Proposition  1.4.  Marshall  (1)  x  -< y i f f - x -< - y ;  (2)  2L  and O l k i n  y i f f -x ^ -y.  (to appear)  use  (2) a s t h e d e f i n i t i o n  of  w, .  But t h e p r o p e r t i e s  appear  i n the  Next  we  o f ™< , b e i n g  dual  t o those  of  seldom  literature.  give  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  a n d weak  maj o r i z a t i o n s .  Proposition  1.5.  (1)  x  <  ) y,  -< u a n d u  (2) Proof  :  smallest > 0.  (1)  Then  < y.  ^ v  o f y.  x ^  Z  ( =-u), v  inequality  f  f  2L  -  ^  J  show  constraints  e  l  that  f  satisfying  f o r which  u satisfies  (2) a r e  yk i s t h e  (x^ +  y_ b y r e p l a c i n g  ... +  x )  y ^ by  x -< u a n d  trivial.  ~* -< ""^J " I i < -Z.  f  u  ... + y n ) -  from  that  (1), a n d  ~y  the index  d=(y^ +  verify  f  o  r  some  u i f f  v.  x v£ y i f f t h e r e  But i n t h i s  i s more  ^  obtained  > y f o r some  can a l s o  The p r o o f  l  of  T  Take  i t i s e a s y .to  < u a n d u .< y .  x^.  e  exists  *k y .  y and x  I f x "w* Z J l  The c o n v e r s e  We x  i f f x ^  component  Also, x  y  (}*> ) y i f f t h e r e  L e t u be t h e v e c t o r  y^-d. u  x ^  ( >  x  case,  involved  we  because  exists  obtain  u  satisfying  u^ by a d d i n g  of the presence  n o t t o be v i o l a t e d .  d^ t o  of the  n  -  D e f i n i t i o n " 1.6.  6 -  A matrix w i t h 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 t o ( l e s s t h a n or equal t o ) 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  of doubly s t o c h a s t i c and s u b s t o c h a s t i c Theorem 1.7matrix  P  (2)  (1)  x  ,  and  i n terms  matrices.  y i f f t h e r e e x i s t s a doubly s t o c h a s t i c  such t h a t x = yP. y i n R^ (x -<Jj y i n R^) i f f t h e r e e x i s t s a doubly  x  substochastic matrix.  P  such t h a t x = y P, where R ( R _ ) i s t h e s e t +  of n o n - n e g a t i v e ( n o n - p o s i t i v e ) numbers. The  ppoof i s o m i t t e d  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 a r e r e f e r r e d t o  M a r s h a l l and O l k i n ( t o a p p e a r ) . The  next theorem i s a p o w e r f u l  t o o l i n m a j o r i z a t i o n , as i t n  reduces many p r o o f s  concerning  m a j o r i z a t i o n from R 1  Theorem 1.8.  2  to R .  h  x ^ y i f f t h e r e e x i s t u , ..., u  such t h a t  - - ] i ^ ^ ^ ' - " < y . - ^ i L y where f o r 1 = 0, 1, . . . , h, u and u "'" d i f f e r i n o n l y two components. x  >  1  =  1+  Proof:  N o t i c e t h a t f o r any v e c t o r z_, z, and 'z_V = ( []_]> •••> z  d i f f e r by a p e r m u t a t i o n which can be r e p r e s e n t e d transpositions. V±  > •••  -  y  n  [n]^  as p r o d u c t s o f  So we. can assume t h a t x^ > ... > x , and  '  Now we proceed by i n d u c t i o n on in  z  n .  I f any o f t h e i n e q u a l i t i e s  (1.1) i s a c t u a l l y an e q u a l i t y , t h e n f o r some  k ,  - 7 -  x.  a  = ( x , . . . ,x ) x  < (y _ . • •,y )  k  x^ = (x, — k+1'  ]  < w •«<•••«< w  J  -<y  = y  k  ,x ) ^ (y. n ^ k+1  5  By t h e i n d u c t i o n a s s u m p t i o n , x  J  x  5  5 J  •< v  , so  ,  a  ,y ) = y n —  ^  ...  .  b  ^ v  .< Z.  a n <  ^  x = ( x , x ) - < ( v , x ) - < . . . ^  (y_ ,x ) <  ( y , x ) -< ( y - , ^ ) -< . . . -<(y_ ,w ) -< (£. ,y_ ) = y_ ,  where two  adjacent vectors d i f f e r  r  If  b  and  of  x ^ x and  their  a  are s t r i c t f c  and we  take  can a p p l y the p r e c e d i n g arguments t o x  6  components i s an  ^ y and  a t l e a s t one  can choose the  u  be  x  inequalities  y_  similarly  and  i n s u c h a way  1  x  and  are  t h a t they  f r o m t h e one  P 6 l y a ( 1 9 5 2 ) , who  original-  considered  components) between  seen i n Chapter  are  y .  i s different  g i v e n by H a r d y , L i t t l e w o o d and  i t will  of the  equality.  proof presented here  But  components.  .  J  o r d e r e d i n t h e same manner as  .  b  fc  t h e 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 and  a  i n e q u a l i t i e s , we  i s i n t e r e s t i n g to note t h a t i f  The  s  i n o n l y two  = (x., +6 ,x„, . . . ,x . ,x -6) 1 '2' ' n-1 n  o r d e r e d , t h e n we  ly  1  { (y +. . . + y ) - ( x . ^ . . . +x i) } > 0  y_ , b e c a u s e  It  all  3  min k=l,...n-l x —  Then 6  b  a l l the i n e q u a l i t i e s 6 =  x  a  2^  :  t h a t t h e above  x proof  c a n be e m p l o y e d t o prove', a more g e n e r a l r e s u l t . i Because we  can r e l a t e  u  i+1 -< u  and  they d i f f e r  them by a d o u b l y  i n o n l y two  s t o c h a s t i c m a t r i x as  components, :  - 8 -  1 u  i  = u  i+1  a  a  a  a •1  f o r s'omeaa.a s a t i s f y i n g form, with  1  i n a l l the  C.  Now  and  The  a+a=l.  A matrix of  e n t r i e s except the  other e n t r i e s except the  two  this  a , and  0  a , corresponds. .%o a -  two  main Idea of Theorem 1.8  can be d e r i v e d  Schur  a>0  i n the d i a g o n a l  "T-transform". then x  a>0,  i s that: I f  x -< y ,  from y by a f i n i t e number of T-transforms.  functions. we  consider  functions  that preserve  majorization. n  D e f i n i t i o n 1.9F:R  (S)*  f o r any  Let S be an a r b i t r a r y subset of R  >• R  i s s a i d . t o be a Schur convex f u n c t i o n  x, y (eS),  concave i f f  -F  It f o l l o w s  x - < y = ^ - F ( x ) < F ( y ) , and  (and  x'  so that The  x,x'  e S ) , then  F(x)  = F(x')  extension  Proposition iff  f o r any  function  (on S) i f f  i s Schur-  from t h i s d e f i n i t i o n that a Schur convex  i s obtained from  if  F  A  i s Schur convex.  Schur concave) f u n c t i o n if  .  1.10. x, y,  (on S) i s n e c e s s a r i l y symmetric, because x  by a permutation of i t s components x  x  1  -< x ,implying  F (x ) <F (x ' ) <F (x ) ,  .  to weak m a j o r i z a t i o n F  (or  i s the  i s Schur convex and x ^  following:  increasing  ( X () y = ^ F ( x ) < F(y)  .  (decreasing)  - 9-  Proof  :  Necessity.  Suppose that  ing.  Then f o r x ^ y , there  So  i t follows  For  s u f f i c i e n c y , n o t i c e that  Hence  that  P i s Schur convex and i n c r e a s -  exists  u  satisfying  x .< u, u < y.  F ( x ) < F ( u ) < F(y) .  F ( x ) < F ( y ) whenever  x ^ y  whenever  x -< y_  x ^ y  of* x < y_  s  or  x < y ,  Therefore  F .'.-  i s Schur convex and i n c r e a s i n g . The  proof f o r the other v e r s i o n of weak m a j o r i z a t i o n The  f o l l o w i n g theorem c h a r a c t e r i z e s  functions  i s similar.  d i f f e r e n t i a b l e Schur  i n terms of t h e i r d e r i v a t i v e s .  Theorem 1.11  (Ostrowski, 1952).  A d i f f e r e n t i a b l e function  F  i s Schur convex i f f I t s a t i s f i e s : (1)  F  (2)  for all.  The  i s symmetric, i , j , ( x . - x . ) ( | f - ~ - ||-) > 0 .  proof i n a s l i g h t l y d i f f e r e n t s e t t i n g 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 verigy  because c o n d i t i o n s  (1) and (2) are u s u a l l y e a s i e r to  than the d i r e c t 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.  I t should be noted that the symmetry property  necessary as can be i l l u s t r a t e d by the f o l l o w i n g  Example 1.12.  Let F(x  1 }  F:R —*-R  be d e f i n e d by  2  x ) = (x -x ) 2  1  = 0  2  i f x-j^ > x otherwise  2  ,  example.  (1) i s  - 10 -  then F  F  Is clifferentiable with derivatives s a t i s f y i n g  i s n o t Schur convex because  ( 2 ) . But  (3/4,1/4) -< (0,1) , b u t  F ( 0 , 1 ) = 0 < 1/4 = F ( 3 / 4 , 1 / 4 ) . Corollary  1.13.  If  n i s d i f f e r e n t i a b l e , then F(x) = J f ( x . ) 1=1 i s convex.  f  1  is  Schur convex i f f  f  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. Schur  convex.  Proof  :  x  l - 2 x  F  i s symmetric and convex, t h e n  F  By Theorem 1 . 8 , we c a n r e d u c e t h e p r o o f t o R .  '  ^1 - ^2 '  x -< y = £ • ( x Hence  If  1 3  x 0 2  FU-^x^  ^  ^  r  e  m  a  r  k  So l e t  f o l l o w i n g Theorem 1 . 8 ,  = X ( y , y ) + A ( y , y ) f o r some X,X > 0, X + X = l . 1  2  2  1  < X P ( y , y ) + XF(y , y ^ ) 1  = XF(  P r o p o s i t i o n 1.15.  e  is  If  f  y ; L  2  2  ,y ) + XF( 2  y ; L  ,y ) = P(y ,y ) . 2  i s convex, then  1  2  n F(x) = If ( x . ) 1=1 1  is  symmetric and convex.  The p r o o f i s s i m p l e a n d i s o m i t t e d . Now we s t a t e t h e t h e o r e m w h i c h w i l l be e x t e n d e d t o t h e s t o c h a s t i c versions i n Chapter 3 • Theorem I . l 6 .  The f o l l o w i n g  conditions are equivalent:  (1)  x -< y ,  (2)  F ( x ) < F ( y ) f o r every Schur convex f u n c t i o n  F  on  n. R ,  - 11 -  (3)  F(x) < F(y)  (4)  n n £ f(x.) < £ f(y.) i=l i=l  f o r every symmetric,convex f u n c t i o n  1  Proof  ( l ) = £ - ( 2 ) , (2)=^(3), (3)=^(4)  :  v  l  y*- (1),  (4 )  "  y  f ( x ) = -x  n n I x. < I y. i=l i=l  x.^v>  x  are convex,  f ( x ) =x  k  k  =  k  +  = max{0,x-y } , k  k n I f(x ) < I f(x 0 i=l i = l ±  k  I  ) = f(y ) = y i = l  ±  k k. I x. < £ y. 1=1 ~ 1=1  that  f(x)=(x-y )  therefore  -Siky  1=1  7V';  and  n  x  < I f(y  Hence  > ... > x  n n n n - £ x < - £ y , so t h a t £ x = £ y . i=l z-i = l i = l i = l  and  n  so  1.15-  a r e c o n v e x , p u t t i n g them i n (4) y i e l d  + ... + x  ±  follow respectively  F i F s t notice that the functions  N e x t we s e e t h a t t h e f u n c t i o n s k=l,...,n  1.14 and  Propositions  we may assume  - ^2 - "•" - n  and  f .  1  f r o m D e f i n i t i o n 1.9, For  f o r every convex f u n c t i o n  F on R ,  ±  for  x  +  + y  k  - ky , k  k=l,...n-1  x ^y The e q u i v a l e n c e  wood a n d P o l y a Mirsky  (1959).  versions  o f (1) and (4) i s p r o v e d by H a r d y ,  Little-  (1929), a n d t h a t o f (1) a n d (3) i s p r o v e d by The e x t e n s i o n  of this  t h e o r e m t o t h e two  o f weak m a j o r i z a t i o n i s d i r e c t a n d  simple.  - 12  Theorem I.17.  The  -  f o l l o w i n g are e q u i v a l e n t :  (1) (2) function (3)  F(x)  F  < P(y)  f o r every  s y m m e t r i c , c o n v e x and i n c r e a s i n g  (decreasing) f u n c t i o n F n I f(x.) <  (4)  n ^ f ( y . ) f o r every  (decreasing) f u n c t i o n The  proof  c o n v e x and i n c r e a s i n g  f .  i s contained i n the previous  repeated. Tomic  ,  The  equivalence  of  (1)  and  d i s c u s s i o n and (4)  i s proved  i s not  by  (1949). 3 Finally  a geometric  p i c t u r e of m a j o r i z a t i o n i n R  In general, f o r a vector z e R  illustrated.  components, t h e r e are s i x permutations T h e s e 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 F i n t h e f i g u r e shown b e l o w . x  l  by  + x  2  z,  + x  3  =  •  K  and  T  figure,  S  I f we  let  S  components.  t h e p o i n t s A,B,C,D,E  They a l l l i e on t h e be  different  and  plane  the set of v e c t o r s  majorized  t h e 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 t h e  o u t s i d e the dotted l i n e s .  S = { x : x -< z)  = ' { x : F ( x ) < F(z_)  T =• { x : x >• _z} = ( x : ( x ) > ( z _ ) p  A  of the  i s t h e c o n v e x h e x a g o n ABCDEF, and  of the plane  If 1  with  is  i s the i n d i c a t o r  p  T  i s that part  Furthermore  f o r every f o r every  S c h u r c o n v e x F} S c h u r c o n v e x F}  f u n c t i o n of the set  A  defined  , . by  - 13 -  l (x) / v  = . 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 Schur  l  +  x  2  lg  i s S c h u r c o n c a v e and  1  T  convex. For  x  that  ,  +  centre  x  3  any S c h u r c o n v e x f u n c t i o n =  K  J  F  increases  F  restricted  to the plane  a l o n g any r a y on t h e p l a n e f r o m t h e  (K/3,K/3,K/3).  Figure.  An i l l u s t r a t i o n  of m a j o r i z a t i o n i n R .  - 14 -  Chapter 2  A.  Introduction. In 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 a n d weak m a j o r i -  zations to the p-majorizations parametrized positive  components.  by a v e c t o r p o f  The H a r d y - L i t t l e w o o d - P o l y a  e x t e n d e d t o a more g e n e r a l f o r m .  inequality i s  The p r o p e r t i e s o f t h e p - m a j o r l -  zations are discussed, with the characterization.by a class of matrices  s i m i l a r t o t h e doubly  p r e s e r v i n g these  TT x  chapter, the parameter p i s a v e c t o r w i t h  components.  i s a permutation > ... > x  Functions  orderings are considered.  Throughout t h i s positive  stochastic matrices.  We s h a l l a d o p t t h e f o l l o w i n g n o t a t i o n : i f of  { 1 , . . . , n} , x e D  means t h a t  : i n o t h e r words , t h e v e c t o r x  77  obtained  p e r m u t i n g t h e components o f x u n d e r TT i s i n d e c r e a s i n g M o r e o v e r , we d e n o t e by x e D i f TT I s t h e i d e n t i t y B.  from  order.  mapping.  The p - m a j o r i z a t i o n s . The  following i s a generalization of majorization.  Definition  2.1.  x ^  y on  D  77  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_ a r e s i m i l a r l y k  ordered),  k for k=l,  n-l  - 15 -  n I P  and  n I p  x  1=1  17  The  extensions  1  71  1 = 1 ^1  1  y  ^1  to x  y and x k y i=l  p l a c i n g t h e a b o v e s y s t e m by n n T p x > J p y . , TT . TT. — 7T.TT. i=k l l i=k I l  n n £ p,x,= £ p.y. )  (or equivalently  f o rk=l,  1  1=1  1  y are obtained k y l i=l  l  1=1  this  and l  I  ..., n.  o r d e r i n g have t o by s i m i l a r l y  ordering writing has  ordered.  be x TT-^D y.  The r i g o r o u s way o f  of the Hardy-Littlewood-  i n e q u a l i t y which corresponds t o t h e case I n which a l l t h e  are equal.  Theorem 2.2. then then  If  n £ p.f(x.) <  1=1  1=1  i f f o r some TT , x ^  n n J p . f ( x . ) < y p.'f(y.) 1 1 — .-,1 1 1=1 1=1 -,  Proof For  n £ p.f(y.)  f o r any s u c h t h a t x_, y_ f o r any TT s u c h t h a t x, y e D  Conversely,  o  Indeed, the  made t h e d e p e n d e n c e o n cTrlce^ea^rncan^u  Polya  .  related  However, t h e r e q u i r e m e n t x, y e D  Theorem 2.2 b e l o w i s a n e x t e n s i o n  p^  TT .  d e p e n d s on t h e p e r m u t a t i o n should  1  by r e -  One r e m a r k a b o u t 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 by  1  :  f o r every  , we h a v e x ^ y,on D  f o r any c o n v e x  77  convex  y on D  subscripts  i  of the general by  .  First  case,  just  TT  , then f .  We o n l y p r o v e t h e c a s e t h a t TT i s t h e i d e n t i t y  the proof  f ,  mapping  replace a l l the  s u p p o s e t h a t x,  y e D and  - 16 -  n n I p ^ f ( x ) - < 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 g e t I p.x. = J p . y . • N e x t c h o o s i n g f ( x ) = (x-y, ) i  1=1  1=1  f o r k = l , ..., n - 1 , we g e t  J P i i " J Pi^k< I P i i=l i = l i = l x  n  f  (  i  x  < j Pi ^i> i = l  }  f  k  < I v r(y ) ±  k  = I  ±  1=1  P f(y ) = ±  k  I P ^ i -I Piy .  ±  k  1=1  1=1  1=1  k k Jp.x.< J p.y. i=l 1=1  so  C o n v e r s e l y , s u p p o s e x A> y o n D.  We may assume t h a t x f ^ y , k D e f i n e A = 0, B =0, A = £ p x , i =l k  k=l,  n.  Q  Q  k  for  k  ±  k ^ . ^ P ^ i Because  a r i d  f  Q  k  =  [  f (  x  k  )  _  f (  y  k  )  ]  7  [ x  i s c o n v e x , we h a v e Q  k  k" k Y  > Q  ]  °  f  k + 1  r k  =  1  »  n  -  -  Thus  T(A -B )(Q -Q )  (A -B )Q  therefore  "fA (Q -Q k=l  < J ^ k=l  therefore  J^\~\zV^  so t h a t  j^k^^k^^yk)^^^^-^-!)-^^-!^^  k  k=l  It  k  k  k  k  k + 1  k+1  ) A Q +  n  K  -  +  n  n  V  B  n  k - l  < 0  n  V  )  Q  W  k  +  ,  V  n  ,  •  c a n be s e e n f r o m t h e p r o o f t h a t t h e c o n v e r s e  <  0  '  i s true f o r  - 17 -  any  This  of the proof However,  part  i s p r o v e d by F u c h s  requires  this  that a l l the p  condition i s left  quoted the s u f f i c i e n t sufficient. for  (1947).  But t h e f i r s t  part  are positive (or negative).  i  o u t by M i t r o n o v i c  (1970), who  c o n d i t i o n o f F u c h s as n e c e s s a r y a n d  I t c a n be p r o v e d t h a t f o r n=2, t h e t h e o r e m  a r b i t r a r y p ^ and p  t h a t i t does n o t h o l d  2 >  holds  Y e t f r o m t h e f o l l o w i n g e x a m p l e we s e e  i n general  i f t h e p^ a r e o f d i f f e r e n t  sign. Example  2.3-  Take p = p = l , 1  p'^=-l, y = x = x = x ^ >  2  1  x, y eeD ; and f o r any c o n v e x < p f(y )+P (y ) P3 ^3)> f  ±  1  +  2  f  b  u  1  y = y t h e n  2  2  f , p-^f (x-^ ) + p f ( x ) + p ^ f ( x ^ ) 2  P x +p x  t  2  1  1  2  2  > p y +P y  2  1  1  2  2  .  F i n a l l y , we r e m a r k t h a t i f x and y a r e n o t s i m i l a r l y ordered,  then the i n e q u a l i t y  v I  H  w  P f(x ) ±  ±  i=l be t r u e o r f a l s e . f(0)+2f(2)  f a l s e because Just  like  v  <  2 P  w f ±  \ can e i t h e r  (y )  c  f  because  a  ±  i=l  Take f o r i n s t a n c e , f ( 5 ) + f ( 3 ) + 2 f ( 1 )  f o r any c o n v e x  f(5)+f(5)+2f(1)  „  < f(6)+  (5,3,1,1) -<(6,2,2,0).  < ( o r >) f ( 6 ) + f ( 0 ) + 2 f ( 2 )  f o r any c o n v e x  (5,5,1,1) and (6,3,3,0) c a n n o t be o r d e r e d the Hardy-Littlewood-Polya  But  f is by-<.  inequality, i t i s  p o s s i b l e t o e x t e n d t h e o r e m 2.2 t o c o n v e x f u n c t i o n s t h a t a r e increasing  ( o r d e c r e a s i n g ) by r e p l a c i n g  The o r d e r i n g d e r i v e d by a f i n i t e T-transforms.  ^>  by -<Jp ( o r - ^ p ) .  i s quite similar to majorization. number o f t r a n s f o r m a t i o n s  I t c a n be  similar to  A l s o , i t c a n be c h a r a c t e r i z e d by m a t r i c e s  similar  - 18 -  to the doubly  stochastic matrices.  Because o f the p r o p e r t y  that x  D, we o n l y h a v e t o c o n s i d e r x  y on D  77  i f fx ^  y  extend  to D  77  y o n D and t h e n  on  77  77  i n t h e o b V i o u s way. Theorem  2.4.  Ifx ^  t h a t x E u° -£> u  Proof  ...  1  i = 0, 1, ..., h , u  y on D , t h e n t h e r e e x i s t  1  u  ^  h  1+  n .  1  h  such  = y on D, where f o r  and u "'~ d i f f e r  : By I n d u c t i o n o n  u ,...,u  i n o n l y two c o m p o n e n t s .  We c o n s i d e r t h e f o l l o w i n g two  cases : (1)  I f any o f t h e i n e q u a l i t i e s  k k J p.x. < J p.y. , k = l , i=l i=l 1  n-l  i s actually  1  1  a n e q u a l i t y , t h e n we c a n w r i t e x  y on  D as : x ,  a  ,  /a a  b  y  a  b /b x . <)  „ on D,  s  where x = (x , x ) ,  y  , a b y = (y , y ) ,  b  ~ on D ,  , a b P = (p , P ) •  N  N  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 , i = l ,  r and  1  w^,  j = l , ..., s  such  that  a ,& 1 /a /a r x < y _ < . . . - £ v x and  b  /a a £ y  ~ o n D ,  /b 1 ,b y-b s ,b b -4? w ^ ...-4} w ^ y  _ o n D ,  s u c h t h a t e a c h two a d j a c e n t  o n l y two c o m p o n e n t s . x = (x ,x ) a  -4 ( y It  b  a  w J  . ) 1  4  order  v e c t o r s above d i f f e r I n  Combining these  (v ,^) 1  -4 • • • ^  i s easy t o v e r i f y  e  4  ... 4  (y ,w ) a  s  (f,x ) 4 b  .4 ( y  that a l l these  and t h a t any two a d j a c e n t  v e c t o r s , we g e t  a  y ) b  3  (y ,x ) a  b  = y on D.  vectors are i n decreasing  vectors d i f f e r  i n o n l y two  - 19 -  components. (2)  I f a l l inequalities  k=l,  6 = . -, and  x  6  =  m  i  are s t r i c t ,  k / J > 0 ,{ I p. ( y . - x . )}  n-1 ^^^I^I i  n  (x + 6/p , x ,  ^ _ ,  i  1  1  2  n  x -4} x  Then F o r t h e n-1  take  y  1  Now  n  "cVp ) • n  on D .  component I n e q u a l i t i e s  must be a n I n e q u a l i t y .  x  of x  ^  y, a t l e a s t  a p p l i c a t i o n o f (1) t o x  one  and y  completes the proof. 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  Ap  be a s e t o f n x n m a t r i c e s  d e f i n e d by  = {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 t h e o r e m c h a r a c t e r i z e s t h e m a t r i c e s Theorem 2.5(1)  eA=e, w h e r e e = ( l ,  A  P P_ 3  Proof eA = e_.  t =  t  in  i  for k=l,  where p^ i s t h e t r a n s p o s e  : Necessity. For k = l ,  components e q u a l  e_, -e e D = ± ^ e A  o f .j4fp.  satisfies:  1); i . e . c o l u m n s o f  k £ a., i s d e c r e a s i n g  (2) (3)  I f A > 0, t h e n A e-«4p i f f A  A  A  sum t o  1 ,  n-1,  o f p.  ^  e, -eA  ^  -e on D ==^>  n-1, l e t z_ be t h e v e c t o r w i t h t h e f i r s t  k  k  to  1  and t h e r e s t  equal  „.k ^ p -•• :i 7,  to  0 .  Then  k k 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 ^ j=l k t k t k n k D — ^ z_ Ap = z p — ^ [ a. .p. = J p. f o r k = l , 1=1 j = l 1=1  z  k  k z_ on  e D D  1 J  1 J  J  1  n.  But  - 20 -  this  set of equalities  Sufficiency.  V k  t o Ap^=jj^.  F o r y e D, l e t x=yA.  F o r i n d i c e s h<k,  = j/hj^j-j/kj^j  x  n-l  This  i s equivalent  m  m  i m p l i e s x e D. yp  t  m  Also, t  = xAp  t = x p - | n  n  P x = ±  k=l,  n - l , applying  £pp y  ±  1=1 For  ±  ±  k  = I  3  1  conditions  k  I  y a  j = l 1=1 k  -  ( 1 ) , (2) a n d ( 3 ) , we  I  a  P  J  a  P  1 J  3  k y  1  y  y  ) k  K  a  i i i p  1 J  +  3  k  j=l  ^  n  1=1  )  y k  K  a 1  1 J  1 J  P  +  y  1  k k = I I i 1=1 J = l y  i - k y  )  a  i i i p  +  p  J  1 J  P  J  + 1  y  n  k I i=l  I i i i j=l a  K  k a 1  i i i  J  p  k  I I j = l 1=1  a  1 J  I i j=l k  k y  ^  Ic  =  1  (  J  k i i j  y K  k  I I ( i j = l 1=1  p J  n  2 I j = l i=k+l  +  J  y a  j = l l=k+l k  k k =11 ^ i j = l 1=1  =  nn  I  k y  k  3  k  +  I I i i i j j = l 1=1 k  =  p  1 J  .  i=l  ky P.x. = k ny y . a . . p . y =i i = i 1=1 J  m  p  1 J  n  1 I k 1=1 j = k + l y  K  a  1 J  i  1  J  P  1  3  obtain  - 21 -  k  k  k  I  1=1 j =L i i i j - j y  k  a  nn XI i i . i i  X  =  y  a  p  1=1 j = l Hence  x  y  ^  J  on  n  I  1=1  j=k+l  k I  =  i  p  p  (1)  D .  i s c l o s e d under  A'  1  £  0 where  p'  multiplication. 0  A'  I  2^  i s the vector  obtained  from  end c o m p o n e n t s ,  matrices  dimensions.  If  x  b  b  of appropiate Z. >  a  X  a  b  where  y  0  A  a  N  r  to out  o  ,  A  o  0  A  n  D  A  b  a  r  p_  such that  e  e Av°  and  • a 0 ^0  (3) o f Theorem 2.5  •  a  a  1  A  £  >-. b  J  properties  (1) turn  u n l i k e t h e case o f doubly  where t h e e x t r e m e p o i n t s  matrices.  a  However, t h e extreme p o i n t s  t o be q u i t e c o m p l i c a t e d ,  permutation  x =y A ,  , then  A  b  by t r u n c a t i n g  1-^,1^ a r e i d e n t i t y  i s c o n v e x , a s c a n be v e r i f i e d u s i n g  stochastic matrices the  b  e  / a b /a bv (x ,x ) = (y ,y )  Atp  conditions  > 4 p possess the following  some l e f t , a n d / o r r i g h t  x =y A  (4)  and t h e e q u i v a l e n t  *<4rp  :  I  (3)  i  i n t h e a b o v e t h e o r e m , we s e e t h a t  properties  (2)  y  1=1  u  From t h e d e f i n i t i o n o f given  y. a. . p .  are simply  -. 22 -  Now  we  come t o t h e 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  of Theorem 2.6. A ej&p For  I f x , y_ e D, t h e n x  such t h a t  the proof  Lemma 2.7.  y on  D  i f f there  exists  x=yA.  we need t h e f o l l o w i n g  Suppose  lemmas. r and l e t a= J p . , i=l  l < r < n , s > t , y>0,  1  i=r+l TT=J-[(s-t)+ (^+|)]  1  Y  ,  then  l l  A  (l)  (s j ...  }  Sj t , ... , t ) — ( s ' j  s'  3  t  ,  f  ) A  I —  aand  A  21  " | ;  A  e  Av  e a c h composed o f i d e n t i c a l  columns, w i t h e n t r i e s  entry  of A  entry  of A^  entry  of A  n entry c=_^p.,  of A  1 1  h  a  v  appropriate  e  i s p.(l-£)/a  th ik  2  2  2 1  for  dimensions; g i v e n by:  1=1,  i s p^n/a  th jk^  22  ;  2 2  th  A  22;  2  1  12  ,  w h e r e A^^ i s r x r , A-^ , A 2 ^ , A  lk  21  A  f o r j = r + l ,...,n ;  i s Pj?/b  th jk (2)  for  where  2  A  2 2  i s p.(l-n)/b  (s+£, . . . ,s+£) = ( + - X s . . . ) A  , composed o f i d e n t i c a l  S  3  3  } S  5  c o l u m n s , w i t h p./c as t h e  -  i k  t  entry, satisfies  h  The  proof  and  i s omitted.  tive  of this  quantity  components sort  lemma  A e>4p.  i s just  The i d e a  from  23 -  a direct  i s that  evenly,  or wholly  o f Jo^ ^ p  Lemma such  2.8. that  Proof  to the f i r s t  I f x e D, t h e n  x=x^A — —  x  ]_ > • ••> that  x n  _ i  k  y from  x  o f u. _-]_;  o f u ^ , by r e p e a t e d  i s chosen  small  (2)  o f Lemma  I f nrY<6< (m+1 ) y j p e r f o r m y  Because we  J then  repeat  of properties  obtained belongs  Now we from  can proceed  definition  of  o f u^.; t h e n  A e  having  so t h a t  a  i twholly  t h e above  i tagain using  t o t h e com-  (1) and (2) o f  tothe  D  .  not exceed Finally,  to the first  procedure  6-my  positive  o f ( 1 ) o f Lemma 2 . 7 ;  i t does to  equal  groups a r e  i ti s distributed  a l lbelong  2.7 t o d i s t r i b u t e  each  distribute  applications  enough  t h e new v e c t o r s f o r m e d  using  some  by an  exists  i n different  t o t h e components  n  groups,  Now  n  and so on u n t i l  k  k  components  and  x-j^.  by  5  into  i . e . x= ( u ^ ,. . . ,u_ , x ) .  y  the multiplication  5  different,  where  component,  s  f o r x^ = ( x , + — , x „ , . . . ,x -, ,x — — ) . — 1 p,' 2 n - l 'n p *± pn  and such  components  posi-  to the larger  f o r a n y 6>0, t h e r e  components  ponents  some  .  : Partition  quantity  verification  can distribute  t h e s m a l l e r t o components  of transformation involving  element  we  algebraic  m  6  apply component  times  i n place  of Y •  the final  matrix  A  tOx*4p.  t o prove  Theorem  2.6.  For necessity,  Sufficiency  we p r o v e  follows  by i n d u c t i o n  on  _  24 -  k n. is  I f any o f t h e i n e q u a l i t i e s actually  an e q u a l i t y ,  o f i^p i s t r u e Otherwise  from  take  J P i  a s i n Lemma  =  m  l  x  ^  x  A £s>4p.  D.  A=A'A"  from  the  following  to R  that  because  FCx ;^) < F(y  T T  orderings we all  functions,  The  by  proof  In  whenever  1T  a r e done.  argument  F  the functions  x =y_A' ,  Hence  ^  y on  the class  we h a v e  D  .  ;p ) f o r x e D D  1 T  =^x  T r  x=yA  .  and $  the  such  extend we  D=^-F(x;p)=  preserves  the involved.  functions,  class  be  Consequently,  of the permutation  o f a l l such  preserve  R  T h e n we  ^ ' " / o n F  that  : D—  :  ) = F ( y ,p) , s o t h a t  of a l l  I f  & the class of decreasing  the following:  preserves  i s similar  case  a n d we  (3)  )} >0.  p, l e t F ( ; p ) x  y on  functions  then  2.9-  of property  p-majorizations.  characterizes  x  ;£  ^p  increasing  Theorem  the  independently  denote  k=l,...,n-l  2 . 8 , x = x ^ A " , A " e >4p •  Lemma  by F ( x ; p ) = F ( x  7 1  i '  y  with  By t h e p r e c e d i n g  : F o r each  F (x ;p )-<F (y ;p)  F(*;p)  .  preserving  orderings  that  D  i  p  ej4p.  Functions  The  see  y on  Also,  !  where  ^  I  k { I p^yj-x i = l  n  k=l,...,n-l Then  i i  assumption  2.8  iuin 6  x  i=l i = l the hypothesis  then  the induction  xo  k  a l l  to that  (%.) i f f F e  o f Schur  convex  of differentiable functions,  we  UpOj ( 9p*0i&0-  function.  can characterize  - 25 -  functions  3p  of  by  their  derivatives.  satisfied : (1) F(x;p) = F C x ^ p ) f o r every permutation TT , T h e o r e m 2.10. If F i s d i f f e r e n t i a b l e , then F e 5"p i f f F (2) (x.-x. ) ( — - — > 0 f o r a l l i , j = l , . . . ,n i " J P § i " Pj § j 7 7  X  M  x  x  ±  Proof  :  First  definition. that  x  e  suppose  To  D  F  prove  .  For  e  ^p,  then  ( 2 ) , assume  6>0  and  (1)  without  i<j  +  +  x  '  = ( X  loss  generality  of  ' ' J-1 P7' J  +  X  +  X  from  +  +  x  +  +  x  +  +  f — i 8  l  P  l  >  (  x  ~  y  x  '  }  x  2  l ~  X  l ^  >  y  2 =  X  +  b  =  3  3  (1)  and  on  D  a  n  .  P l  d  2  2  hold.  We  Because  x  p (y -x ) 2  fi  (2)  y^Cy^yp)?  2  l -  3  p  a  ^  ,x ),  x  b  3  +  i ^ p  suppose  preserves  y  =  x  have  X l  ' n  E(x p)-F(x' F(x ;p)-F(x';p) lim ^ •— , — = lim _. .,_ — ;> 6->0 -V5/P, j 6+ 0 6/p aw i aw F(x ;p)-F(x ;p) i | _ _ 1_ lim, ~ ~ > 0 . • x p x + i  x=(  >•••  X  a  Next  x  +  I P T P 7 ' • • > i - i p T p T > i p T > • • • > V I P 7 ' J •' • • > n> •  Taking  we  the  , let  ^ V p T + p T ' • • • ' i pT p7' i l pT'• x  follows  2L l  +  Z  P 2 >  of  0  x  2 .  =  o  p  l  y  For  need  Theorem  n  l  only  D  +  P  a  2  y  n  2  some  2.5  to  prove  , we  may  take  we  have  T h e r e f o r e we  have  2L^y • > t h e n  d  ' e  that  , 0.<6<1,  F  - 26 -  P(x  ; E  )-P(  where " j ^ " Hence  I ; E  )  {(y^ff-)^ (y -x )p-jj,  = -  i n d i c a t e s evaluation at the point  F ( x ; p ) - F ( y ;p_) = 2.11.  Corollary  If  F(x;p)= J p f ( x ) i  Just  1  like  hyperplane  i s differentiable,  p^x^+...+p x =K  n n I p.,...,K/ 1 p.) 1=1 i=l  order  Q  1  i s convex.  restricted  to the  a t t a i n s i t s minimum a t t h e p o i n t  w h e r e a l l components a r e e q u a l .  In  1  t o see t h i s , i t i s s u f f i c i e n t lying  77  f  5 °-  then  S c h u r c o n v e x f u n c t i o n s , F e 3p  1  y £ D  f  1  (l-9)x+9y  j f-J^ - ffj|j  (y -x )  P i  rr belongs to J p i f f  n  z=(K/  2  2  t o show t h a t any p o i n t  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_ ^  W i 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.  y_ on D . 77  I f f o r some k,  k=l,...,n-l, P-LY-L+. • - + P y k  k  <  P  — — + . •- + P  l  I ?l  k  1=1 then  P  l  y  k  +  . .. p y +  k  k  ^—  1=1  < p ^ t . . .+P y k  k  < P^-^—  I  1=1 So we h a v e  +• • - P — +  K  k  P  I  Pi 1=1  ±  n y <...<y,<K/ £ p. . 1=1 n _  Therefore  ,  IP i  p  _  k  p.., y , . + . . . + p v k+l^k+l*n°n n  1  < p ^k+1 n  + • • • +P *n n  v + 1  I  1=1  P  ±  I  1=1  K  P  ±  - 27 -  Summing t h e f i r s t  and t h e l a s t  n J P-Y- < K . 1= 1  contradiction  1  on D . any  to  on t h e h y p e r p l a n e  includes the class t h e case  r  that  z_ ^  P  i n c r e a s e s from  P]_ ]_ • • • P x  +  +  n  x n  =  z_  ^ '  along ^  e  c  -'-  ass  p, = ... = p l ^n  y  l e t us c o n s i d e r s t r a t i f i e d  s t r a t a , and w i t h  subpopulations.  N^,...N  as t h e s i z e s  sampling  of the  Suppose t h a t t h e v a r i a n c e w i t h i n e a c h  stratum  2 is  y  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  As a n i l l u s t r a t i o n , with  Hence we c o n c l u d e  1  Therefore, the function  ray lying  3p  i n e q u a l i t i e s , we a r r i v e a t t h e  known ( s a y , e s t i m a t e d f r o m p a s t e x p e r i e n c e ) a s  2  S-^,...VS  .  I f we draw s a m p l e s n^,...,n from t h e c o r r e s p o n d i n g s t r a t a w i t h s a m p l e means y ^ , - - - , ^ , t h e n t h e e s t i m a t e o f t h e p o p u l a t i o n mean i s  y =  ^ N.y./N i=l 1  V  , where  N = J N. ; w i t h v a r i a n c e i=l  1  r r = ( 1 / N ) I N . ( N . - n . ) S ? / n . = ( 1 / N ) £ N?S?/n. - c o n s t a n t . — 1=1 1=1 2  For f i x e d  2  sampling  to  choose s a m p l i n g  or  equivalently  x. = n./cT/N.S. I 1 1 1 1 Since  f  cost  c  ]_ ]_ • •• rv n  units  t h e sum and  +  n  +C  n  so as t o m i n i m i z e  2 2 J N.S./n. 3_ 1 1 1=1 •  ^ ' ^  =  - i  '  . •  With —  v  f ( x ) = 1/x , we h a v e  i s c o n v e x , t h e minimum o f  e  °bJ  e C T J  i  v e  the, v a r i a n c e ,  p. = N . S ./c~7 1 1 1 1  '  is  ,  2„2 N7S7/n. = p . f ( x . ) i i i ^ 1 1  r £ p.f( i=l  x ±  ) Isattained  - 28  where  x, = 1  agreeing  ...  = x  , i.e. r  5  w i t h the r e s u l t  -  n /c"T/N, S,. = ... 1 1 1 1  = n /c /N S_ r r r r ,  n  obtained  u s i n g the Lagrange  multiplier  technique. One partial  a d v a n t a g e o f u s i n g t h i s a p p r o a c h i s t h a t we orderings  V(x) to  the  < V(x')  , where V ( x )  sampling units  because  f  by  x'  x %  d e f i n e d on  n  x  such t h a t  ^  i s the v a r i a n c e  associated with  i s d e c r e a s i n g , we  x  can  x  strengthen  on D ^ = = > V ( x ) < V ( x ' ) .  .  x'  have o n  ^  corresponding Furthermore,  the above  result  -  29  -  Chapter A.  3  Introduction. In t h i s  chapter,  stochastic versions deal with  we e x t e n d t h e m a j o r i z a t i o n s  and d i s c u s s  to their  t h e r e l a t i o n s among them.  t h e p r e s e r v a t i o n t h e o r e m o f S c h u r c o n v e x i t y by  P r o s c h a n a n d S e t h u r a m a n ( 1 9 7 7 ) and o f m o n o t o n i c i t y , a class of inequalities distributions.  hypothesis  t h e p r e s e r v a t i o n o f Schur  a class of test functions  f o l l o w i n g notions  Definition  3.1-  (1)  X  o f s t o c h a s t i c o r d e r i n g between  i s stochastically  less than  Y , de-  , P r {X>%}<Pr {Y>t },  t  P (t)<F__(t). X = S The  i f f X st<Y  and  Y s t < X.  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.  If  X s t < Y , t h e n EX<EY., p r o v i d e d  t h e expecta-  tions are f i n i t e . Proof  real  repeatedly..  n o t e d by X stjc Y , i f f f o r e v e r y r e a l  (2)  i n one t y p e o f  of majorizations.  random v a r i a b l e s a r e u s e d  i.e.  con-  testing.  Stochastic versions The  obtaining  t h a t a r i s e i n some m u l t i v a r i a t e  Then we a p p l y  v e x i t y 'to c o n s t r u c t  B.  We  :  EX=  rO  - P r { X < t } d u ( t ) +• o  •'0  Pr{X>t}dy(t)  - 30 -  and  ° Pr{Y>t}dy(t) ,  -Pr{Y<t}dy(t) +  EY =  0  where y s t a n d s f o r t h e L e b e s q u e m e a s u r e or* t h e c o u n t i n g The  inequality  EX < EY  the  corresponding  measure.  now f o l l o w s f r o m t h e i n e q u a l i t i e s o f  integrands.  I n t h e c a s e o f d e g e n e r a t e random v a r i b l e s , t h e s t o c h a s t i c inequality  reduces t o ordinary  S i m i l a r l y , various are  inequality.  stochastic extensions  of majorization  p o s s i b l e w h i c h , f o r d e g e n e r a t e random v e c t o r s , r e d u c e t o  ordinary  majorization.  authors,  are obtained  These e x t e n s i o n s , from t h e f o u r  p r o p o s e d by v a r i o u s  equivalent  Theorem 1.16 by c h a n g i n g t h e c o n s t a n t s  conditions of  t o random v a r i a b l e s a n d  r e p l a c i n g t h e i n e q u a l i t i e s e i t h e r by t h o s e o f e x p e c t a t i o n s , o r by  stochastic inequalities.  However, t h e s e v e r s i o n s  t o be q u i t e d i f f e r e n t e x c e p t u n d e r c e r t a i n i m p o s e d some o r a l l o f them become e q u i v a l e n t . M  X  l  Y with  F(X)  M  2  M M  4  V  :  conditions,  These v e r s i o n s a r e :  probability 1 .  < F ( Y ) w i t h p r o b a b i l i t y 1, f o r e v e r y S c h u r c o n v e x  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)  3  t u r n out  < E F ( Y ) f o r every symmetric convex  E {f ( X ) + . . .+f (X )} 1  j/Ei]  -  st<  j/t-i]  < E { f ( Y ) + . . . + f (Y 1  f o rk=1  --- n  1a n d  where X , >...>X -, i s t h e r e v e r s e d [1]- [n] r  n  r  )}  F . f o r every convex f,  j/ci] order  F.  =j / c i ]  statistic.  - 31 -  M  *  n  n  I Y i=k  :  L  M:  (EX  6  r i 1  I X i=k  tst<  [ 1 ] 3  n  ...,EX  [ n ]  f  -, f o r k = l . . . . , n - l a n d  )  ^(EY  [  i  r  ... EY 3  [  n  ]  n  £ Y  1=1  =  m  by  slight modification.  ^ (-«( ) a n d 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 ) e x c e p t f o r v e r s i o n 5-  resulting versions  obtained  v e r s i o n 5, we h a v e : k k P r-: y Xr-.-, s t < 7 Y .->  V  Y  i=k  s t  i I ru x  i=k  L 1 J  we r e p l a c e •< functions i n  We s h a l l d e n o t e t h e (Q-^, • . . ,Qg) .  For  f o r  k=1  >--->  these versions  n  L 1 J  The f o l l o w i n g two r e s u l t s  concerning  t h e r e l a t i o n s among  a r e due t o M a r s h a l l a n d O l k i n ( t o a p p e a r ) .  t Proposition Proof  ^  n  I rii  5  by P^,...,Pg  and  f o r k=l,...,n .  r  n  *w*  F o r st-< ( s t ^ O , w  the above v e r s i o n s  r  ).  We c a n 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 similarly with  J X  1=1  3*3-  !  a n d M^ a r e e q u i v a l e n t , so a r e M^ a n d  : The e q u i v a l e n c e  of  M  2  i  a n d M-^ f o l l o w s f r o m t h a t o f  x -< y and F ( x ) < F ( y ) f o r a l l S c h u r c o n v e x f u n c t i o n s . F . i  M  2  Now  ==^- M  2  f o l l o w s from P r o p o s i t i o n  suppose M  holds.  2  3«2.  For arbitrary  t  and Schur  function  F , l e t S = { z_: F (z) >t}  .  Therefore  Pr{F(X)>t} = E l ( X ) < E 1  (Y) = P r { F ( Y ) > t } .  t  Hence M =^-M, 2  s  Then  lg  convex  i s Schur  convex  - 32 -  M  Theorem 3-4.  1  ^  2,_:  S M,  5  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 . F o r t h e p r o o f and. t h e c o u n t e r e x a m p l e s , (to  appear).  Proposition 3• 5• and  see M a r s h a l l and O l k i n  I f X^,...,X  are i . i . d . ,  n  X,+...+X^ = Y-. + ...+Y , t h e n 1 n. 1 n'  X ,... X 1  J  are i . i . d . ,  n  Y  1 3  Y^...^  X. = Y. . i  are i . i . d .  Inparticular,i f i ^ '  ...,Y are i . i . d . ,  t h e n M-^M^M^ a n d M,_  n  are e q u i v a l e n t . Proof  :  Denote t h e d i s t r i b u t i o n s  X ...+X 1 +  where  =" Y + . . . + Y = >  n  1  n  [$ 0  n  o f X.,Y. by l l 5  J  = [$ ] =>$ n  x  y  x  a n d $, . Y  X  r  = $ = > *X ~ ^Y ' y  $' i s t h e 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^ ,M ,M,_ ,M,_ r e s u l t s 2  from :  M = > M^^r^M,- o r M* r=b X + . . .+X 1 r 2 7 5 5 1 n n  n  r  It  may a p p e a r t h a t M,- a n d M  equivalence For obtained. previous  r e q u i r i n g such  = Y + ...+Y . 1 n n  areclosely related,  strong conditions i s a b i t surprising,  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 The p r o o f s a r e e a s y e x t e n s i o n s o f t h o s e results.  so t h e i r  c a n be  of the three  - 33 -  P r o p o s i t i o n 3-6. P ( Q ) and 2  2  a  n  d  a  r  2  q i u  v  a  -l  e  n  t  >  s  o  a  r  e  2  p P  p  l = > 2 .  \  P  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  t h e P's by t h e Q's.  P r o p o s i t i o n 3.8. and  e  P (Q ).  Theorem 3.7.  replace  e  X +...+X 1  n  I f X^,...,X  are i . i . d . ,  s t < Y-j^t. . .+Y , t h e n X v  n  1  Y-^,...,Y  s t < Y- .  n  are i . i . d . ,  In particular, i f  L  X-,,...,X a r e i . i . d . , Y-,,...,Y a r e i . i . d . , t h e n P , P and 1' ' n ' 1 ' n ' 1' 2 (Q-|_,Q and Q^) a r e e q u i v a l e n t . n  n  5  P 5 r  2  Because v e r s i o n 1 i s t o o 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 o f i n t e r e s t , we s h a l l u s e v e r s i o n 2 as o u r d e f i n i t i o n for  s t o c h a s t i c m a j o r i z a t i o n a n d weak m a j o r i z a t i o n s  P r o s c h a n a n d S e t h u r a m a n 1977)-  (as i n N e v i u s ,  Prom now o n , 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 n o t e x p l i c i t l y s p e c i f i e d . Version  3 has t h e f o l l o w i n g  P r o p o s i t i o n 3-9and  u -< v , t h e n — ^ —'  o f v e r s i o n 3.  If.X^,...,X  illustration:  a r e e x c h a n g e a b l e random v a r i a b l e s  (u,X, ,. . . ,u X ) st-< ( v , X , , . . . , v X ) i n t h e s e n s e 11' ' n n' 11' ' n n w, ) a n d The same i s t r u e i f we r e p l a c e «< by (\ w* (  s t < by st-£ (st\ )• Proof  : I t s u f f i c e s t o p r o v e t h e c a s e n=2 b e c a u s e o f Theorem 1,  In t h i s  case,  (u^,u ) 2  "4. (  V  ]_J 2^ — — ( u - ^ U g ) v  =a (v-^ , v )+cT( v , v-^) 2  2  -  for  some a,a>0, a+a=l-  F o r any  EF(u X ,u X ) = 1  1  =  2  34  -  symmetric  convex  function  F  ,  EF[( v +aV )X ,(aV +av )X ]  2  a  EF[a(v X 1  1 3  1  2  1  1  2  2  v X )+a(v X ,v X )] 2  2  2  1  1  2  < a E P ( v X , v X ) + a E P ( v X , v X ) = EF ( v ^ , v X ). 1  The  1  2  p r o o f f o r t h e P,Q  2  2  1  1  2  2  versions are s i m i l a r ,  2  except w i t h the  a d d i t i o n of monotonicity. The  M^  v e r s i o n o f t h e above p r o p o s i t i o n i s p r o v e d  M a r s h a l l and  Proschan  that  e x t e n s i o n t o M _,is n o t p o s s i b l e .  further  p r o v e d by  also i l l u s t r a t e  2  Chong  by a n  The  P^  example  version i s  (1976).  Intuitively, an i l l u s t r a t i o n , parameters  ( 1 9 6 5 ) , who  by  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. .  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  N, p where p > 1/2,  then X tends t o take  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 ] ; o r e q u i v a lently,  X t e n d s t o be g r e a t e r t h a n N-X.  Consequently,  (X+c,N-X) t e n d s t o m a j o r i z e (X,N-X+c) f o r any for  any  Schur  convex  EF(X+c ,N'--X) =  so  function  c>0.  In  fact,  F ,  N I P(x+c ,N-x)Pr{X=x} = x=0  N J F (N-x+c ,x )Pr{X=N-x} x=0  EF(X+c,N-X) = i  Similarly,  I [F(x+c,N-x)Pr{X=x}+F(N-x+c,x)Pr{X=N-x}] x=0  .  (1)  EF(X,N-X+c)  =i d  N I [F(x,N-x+c)Pr(X=x}+F(N-x,x+c)Pr{X=N-x}] x=0  (2)  - 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 e a c h summand i n (1) i s g r e a t e r §n<B  i n (2).  Hence  t h a t F(X,N-X+c) In p a r t i c u l a r ,  ,  >  than the corresponding  (X,N-X-c )st-< ( X + c , N - X ) , w h i c h a l s o means  s t < F ( X + c , N - x ) f o r any S c h u r c o n v e x i f f  f u n c t i o n F.  i s c o n v e x , t h e n F ( x ^ , x ) =f (x-^ ) + f ( x ^ ) i s 2  S c h u r c o n v e x , and f ( X ) + f ( N - X + c ) st<  f(X+c)+f(N-X)  ,  w h i c h i s a lemma p r o v e d by Cohen and S a c k r o w i t z (1975)• C.  Preservation  o f Schur  convexity.  Now we come t o t h e p r e s e r v a t i o n F i r s t we i n t r o d u c e Definition totally  two  functions,  definitions.  F o r A C R,  3.10.  theorem o f Schur  IJJ(X,X):AXR—*-R  i s s a i d t o be  p o s i t i v e of order 2 (TP ) i f 2  (1)  i|>U,x) > 0,  (2)  f o r X-j^jXgeA, A ^ X 1  2  and 0 < x < x , 1  2  ^(X ,x )i|j(X ,x ) > iJ;(X ,x )i(;(X ,x ) . 1  Definition  3-H-  1  2  2  I f ACR  1  2  2  1  i s s u c h t h a t X-j^, X e A , X-j^>X 2  2  X ±X eA , 1  2  then 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 x s O , (2)  f o r any £ , X e A ,T\> ( X + X ,x) = 2  1  2  i|» (\ ,x-y )ty ( X ,y ) dy (y) , 1  2  - 36 -  where  y i s Lebesque measure  or counting  measure,  Prom now o n , we s h a l l o m i t t h e dummy i n d e x i n t h e s u m m a t i o n I  a n d t h e p r o d u c t ~J~|" w h e n e v e r  following  I t i s c l e a r from t h e context.  The  i s t h e 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  negative function function  F(x)B(£  If  Xl  JX JrFr ( X , x ) d i i ( x . ) . . .dy ( x ) ±  S c h u r c o n v e x i n A_ , where  Proof  t h e n f o r any Schur convex  F , H(A_) d e f i n e d by  H(X) = is  o f two v a r i a b l e s ,  a n d SGP, B i s a n o n -  2  1  i  1  n  t h e i n t e g r a l i s assumed  to exist.  : I t s u f f i c e s t o p r o v e t h e c a s e n=2 b e c a u s e o f Theorem 1.8  (X£,X ) < ( x j ,  X ) , then X-^X^xj+X^. 2  2  t  Hence G ( x is  1 3  x ) = F(x^,x )B(x^+x ,X-^+X ) 2  2  2  i  = F (x , x )B ( x + x , X- +X ) ±  2  2  1  L  2  Schur convex i n ( x ^ , x ) 2  Because  G  i s s y m m e t r i c , i t i s easy t o v e r i f y t h a t t  H  i s also  t  symmetric.  T h e r e f o r e we may assume t h a t X >X-^>X >X  Now H(X-j_,  2  1  2  .  2  X )-H(X ,X ) I T G ( x - , x ) [ i p ( X , x ) i J ; ( X , x ) - i J ; ( X , x ) i J j ( X , x ) ]dy ( x ) d y ( x ) 1  1  2  2  1  1  2  2  1  1  2  2  1  2  t  G(x  1 5  x ) L>(X  >^ -Y )ty ( X , x ) - i j j ( X , x ) i ^ ( X , x - y ) ]  2  1  2  2  1  1  2  2  t  xijj ( X - ^ A ^ y ) d y ( x ) d y ( x ) d y ( y ) 1  ty ( X - X , y ) 1  [iJ;(X ,x. )^(X ,x )-i (X ,x )i (X .x )]  1  x-^>x  2  1  1  2  2  r  1  2  r  2  1  2  x[G(x +yix )-G(x ,x +y)]dy(x )dy(x ) 1  2  1  2  ±  2  2  - 37 -  >  0  •>  where t h e s e c o n d e q u a l i t y the t h i r d  follows  f r o m b e i n g SGP a n d  \^-\^=\2'  equality  follows  f r o m c a n c e l l i n g t h e p a r t x-^=x^,  changing v a r i a b l e s  and G  being symmetric; t h e l a s t  follows  TP2J  from b e i n g  G  inequality  Schur convex and x ^ > X 2 , r e s u l t i n g i n  the integrand being non-negative. Because  F  i s Schur convex i f f  -F i s S c h u r  concave,  Theorem 3 . 3 2 i s a l s o t r u e i f we r e p l a c e S c h u r c o n v e x i t y by S c h u r concavity. F(x-^,x^)=f  convex  Furthermore, from Mudholkar's r e s u l t (x-^)f (x^)  i s Schur convex  (log-concave).  Corollary  3.13-  (concave) I f f f  So we h a v e t h e f o l l o w i n g  ( P r o s c h a n a n d S e t h u r a m a n 1974).  TP^ a n d SGP a n d f  (1966), i s log-  corollary: If-tifi (.A ,,x) :.is  i s log-convex (log-concave), then h(X) \p ( X ,x )g (x )dy ( x ) i s l o g - c o n v e x ( l o g - c o n c a v e ) .  d e f i n e d by h(X). = We g i v e b e l o w  some commonly e n c o u n t e r e d f u n c t i o n s t h a t a r e  w e l l known t o be T P ^ a n d SGP. P r o p o s i t i o n 3-14.  Verification will  The f o l l o w i n g  ty(X,x)  be o m i t t e d .  a r e TP^ a n d SGP:  X  x  (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,...,  =  h e r e we a d o p t t h e c o n v e n t i o n t h a t  (3)  T|>U,X)  vlxj  = ^ y - y  f o r X>0 ,  A>0,  A=l,2,..., ( ) 0 i f x>X, =  y  X''  - 38 -  T  =x  \r\l)f o r x = 0,.l,2,... ,  (4)  ip(\,x)  with  "ijj(x,x)=0 o t h e r w i s e " b e i n g assumed.  a) (x;X. ) = c B ( , j ^ X ^ ) 7 T ^  In case that a random v e c t o r  X>0,  X A  5  ( X ^ jX^ ) i s t h e d e n s i t y o f  H(X.) i s t h e e x p e c t a t i o n  E (F(X)). X_  Theorem  3-12  t  X_ = ^  c a n be w r i t t e n i n t h e f o r m X^  c e , we i n t r o d u c e Definition functions  X  st-< X , . X X_  the following:  3-15.  A family  of n-dimensional d i s t r i b u t i o n  { $ , : X e A } ( o r random v e c t o r s  { X : X e A } ) i s s a i d t o be  n  n  A  A_A  a Schur convex  For convenien-  A_  family  ( i n X) i f H(X)=E  (F(X))  i s Schur convex i n  A  _X w h e n e v e r  F  i s Schur  Proposition  3-1^".  convex.  {$,} i s a S c h u r c o n v e x  family  A  X  i f X =(X , A. ]]_  ) i s composed o f i n d e p e n d e n t c o m p o n e n t s o f t h e same  family,  n  and X  1=1,...,n h a s d e n s i t y i families:  from e i t h e r o f t h e f o l l o w i n g  A  (a)  Poisson  (Rinott  1973):  X  -X  X  <S(x;X) = (b)  Binomial  f o r x = 0,l,2,...,  (Nevius,  P r o s c h a n and Sethuraman  $(x;A) = ( ) 0 ( 1 - 8 ) A  and (c)  X>0 .  X  A _ X  1977):  f o r x=0,l,2,..., X = l , 2 , . . . ,  f i x e d 9>0.  Gamma ( N e v i u s ,  P r o s c h a n and Sethuraman  1977):  - 39 -  cj>.(x,A) Proof  : This  =  Q X T  ()  E  6  X  F  O  R  X  °  >  X  i s because  a n d f i x e d e>0.  A>0  3  jy<|>(x  thej o i n t density  A^)i  s  cB ( J x ^ , £ A ^ )"TT^ (  > i ^ w i t h ip i n t h e f o r m 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 S c h u r  x  convex  family  A  has d e n s i t y  A^  in the following  form:  (1)  ( R i n o t t 1973):  Multinomial  I fX  , x. <|>(x;X) =  NITJ-^  x. l  for  (2)  x = 0 , l , . . . ,N ±  Dirichlet  i  Inverted  ±  (Nevius,  for x >0, J x < l , (3)  £x =N,  3  1  A >0,  l\ =l.  1  ±  P r o s c h a n a n d S e t h u r a m a n 1977):  A > 0 , a n d f i x e d 3>0. 1  Dirichlet  (Hollander,  Proschan and Sethuraman, t o  appear): r(e+£A,) r ( e ) ( i  +  (4)  1  Negative multinomial  x  I x . )  f o r x > 0 , A >0, and f i x e d 1  x , i e  ^ i "  T  ^  r  y  ( M a r s h a l l and O l l t i n , t o appear)  A .  A  x.  i  —d-I^TT^—  = f(k)  1  9>0.  r(k+Ix ) <t>(x;A)  +  _  1  x,!  -  40 -  f o r x = 0 , l , 2 , . . . , Xj_>0, J x < l . a n d 1  f i x e d k>0.  1  Multivariate  negative  binomial  (Nevius, Proschan and  S e t h u r a m a n 1977) :  r(N)  x ! ±  f o r x = 0 , l , 2 . . . . , A >0, a n d f i x e d N>0. 1  Dirichlet  ±  compound n e g a t i v e  multinomial  (Hollander,  Proschan and Sethuraman, t o appear):  <Kx;X) =  r(N+J x,)r(e+yx,)r(N+e)  r(x.+A,)  i i TT r(N)r(e)r(N+e+^A +^x ) x1:r(x1) 1  i  f o r x = 0 , l , 2 , . . . , A >0; a n d f i x e d N>0, @.>0. i  ±  Multivariate  hypergeometric  (Nevius, Proschan and  S e t h u r a m a n 1977):  i f o r x = 0 , l , . . . , },'x =N, A = l , 2 , . . . , J A ^ M ; a n d f i x e d i  1  i  positive  i n t e g e r s M, N w i t h M>N. Multivariate  inverse  hypergeometric: -1 V / M - j A .\  M ^k+yx.-l*  7  ^ 1 f o r x.=0,1,2,...,X  " M§  M-£ A .-k+1  14  <f>(x,A) =  1  #X .  x  * k - l M - y x . - k + l ' ' Vx J lx./ 1 '1 1  L  =1,2,... , J X ^ M - k ; a n d f i x e d  positive  i n t e g e r s M,k w i t h k<M. Negative  multivariate  and S e t h u r a m a n 1977):  hypergeometric  (Nevius,  Proschan  -  r(x.,+x,)  • N.!.r(M)  d,'(xA) = , 3  r(N+M)  f o r x =.0,l,2, . 1  TT———  x.!r(A)) 1 1  . . , Jx =N,  A > 0 , ^A =M; a n d f i x e d M>0, a n d  ±  positive integer  i  ±  N .  The p r o o f f o l l o w s d i r e c t l y The d i s t r i b u t i o n s because  41 -  f r o m Theorem 3-32 a n d P r o p o s i t i o n 3'. 14  ( 1 ) , (7) a n d (9) a r e ( n - 1 ) - d i m e n s i o n a l  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^+...+x =N. n  P r o p o s i t i o n 3.18.  (1)  I f {$,} i s a S c h u r c o n v e x  family,  A  t h e n Pr{r- < (or< )X^< (or< ) r L  2  f o r a l l i|A} i s a Schur  f u n c t i o n o f A_, where r-^>-oo  }  (2)  concave  r |<». 2  I n a d d i t i o n , i f X^ i s n o n - - n e g a t i v e a n d Z A  i s t h e number o f  A_  z e r o c o m p o n e n t s o f X.. , t h e n A -< A' = ^ Z st<Z , ~~A A A. —  Proof  : (1)  I t i s easy t o v e r i f y (1  P(x)  is  i f m i n { x . }> (or> ) r a n d max.{x. }< (or< ) r i " otherwise 3  1  =\  0  Schur concave.  t h a t F ( x ) d e f i n e d by  1  1  _  0  1  Therefore  E^F(X) = P r { r < ( o r < ) X < ( o r < ) r f o r a l l i|A} 1  is (2)  Schur concave  i  2  i n \.  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 o f {x>0:. 1% which a r e Schur convex on  i  = 0 )  <k}  H> 0,  - 42 -  The  multinomial  c a s e o f (1) w i t h r =°° c o r r e s p o n d s t o t h e 2  result  o f O l k i n (1972) a n d w i t h r =-<» c o r r e s p o n d s t o t h a t o f  Rinott  (1973)-  result  o f Wong a n d Yue (1973).  D.  1  The m u l t i n o m i a l  c a s e o f (2) c o r r e s p o n d s t o t h e  Preservation of monotonicity. F o r 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' f o r every  l  strfr ( s t ^ ) Xx , <  x  >  E F(X) < E ,F(X) A  Schur convex i n c r e a s i n g ( d e c r e a s i n g )  X  F.  Thus, i n  a d d i t i o n t o t h e p r e s e r v a t i o n o f S c h u r c o n v e x i t y , we need a preservation of monotonicity Theorem 3.£9»  as w e l l .  L e t cfj (x ;X_) be a d e n s i t y o f t h e f o r m cj)(x,X) = c B ( [ x , J X ) T T ^ ( X , x . ) , 1  i  where y i s SGP a n d  B  i  satisfies:  B(x,X) =  B(x+y,X+a)^(a,y)dy(y) ,  w h e n e v e r a,XeA, a>0. Then t h e t r a n s f o r m a t i o n F — H H  preserves then Proof is  iA)  =  d e f i n e d by  ••• F ( x ) * U J I ) U (*-,_). . . d y ( x ) d  n  J  J  monotonicity,  i.e. i f F  i s increasing or decreasing,  so i s H . : Suppose  F  i s increasing.  i n c r e a s i n g i n i t s k^  F o r a , X e A a n d a>0, i  b  We o n l y need t o show t h a t  component f o r k = l , . . . , n .  H  - 43 -  H(X-|_> • • • j A =c  + k  a , . . . ,X ) n  F(x)B(^x  =c  jA +a)m(X  ±  1  x ) j (A +a x )JTd (x )  ± 3  1  F(x)B(£x. J X . + a ) T T ^ ^ i^k  F(x ,  . .. ,x +y, .  1  k  1 3  1  J  k  x )^(x 1  3  y  1  x -y)^(a y)TT Vi^ i) y^) d  k J  k  . . ,x )B( Jx.+y ±  n  k  x  d  3  ,^X +a)TT^(X,x )4)(a,y) 1  1  x^pdy (x1)dy(y) F(x)B(jx +y j x + a ) T T ^ ^ ' i H ( a y ) T T y  >c  x  ±  i  d  i  F(x)B(Jx.,Jx )TT^(A x.)TTdy(x.) 1  equality  equality  from a change o f v a r i a b l e ;  follows  from  F  follows  a n d y>0  follows  The c o n d i t i o n t o be s a t i s f i e d marginal  density  i s i n a similar  ) d  vi(y)  f r o m t h e SGP; t h e t h i r d  being increasing  y<0); a n d t h e n e x t e q u a l i t y  i  = H(X) ,  1 3  where t h e second follows  ( x  5  the next  inequality 0  for  from t h e h y p o t h e s i s on  B .  by  (integrand  B  i s simply  i s  that the  f a m i l y , only w i t h a lower  dimension; n i.e.  i f  (|>x 1  x  XT...X 1 n n  (x, ,...,x ) = 1  n  n  cB( £ x , Y 1=1 1=1 1  n-l then  <f>x-,.,,x X'1...X n~,  1  In p a r t i c u l a r ,  (x ,. . . ,x  n-l  1  n  1  ) ,  1=1  1=1  n-l  [ L ) 1  JWx^x ) .  1= 1  i f the m u l t i v a r i a t e density i s a product of  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 density  1= 1  \ ) TT^(X.,x 1  n-l  ) = cB( [ x 1  n  i s of the form  SGP, t h e n t h e m u l t i v a r i a t e  ( X^ , x^ ) , so i t p r e s e r v e s  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 hypothesis  monotonicity,  o f t h e above  - 44 -  theorem,  so w i l l  I t smarginal densities.  P r o p o s i t i o n 3.20.  The S c h u r c o n v e x f a m i l i e s o f i n d e p e n d e n t  P o i s s o n , B i n o m i a l a n d Gamma o f P r o p o s i t i o n 3-26 p r e s e r v e monotonicity. The p r o o f f o l l o w s f r o m t h e f o l l o w i n g o b v i o u s lemma: Lemma 3.21.  I f ip(X,x) i s T P ( o r S G P ) , t h e n s o a r e a ty(X,x) a n d 2  b^i|j(A,x) where a,b>0. Now we g i v e some e x a m p l e s  of multivariate  that s a t i s f y  t h e h y p o t h e s i s o f Theorem 3.19-  ly different  from t h e i r p r e v i o u s forms  b e c a u s e we r e q u i r e t h e d i s t r i b u t i o n s (1)  distributions Some a p p e a r  shown i n P r o p o s i t i o n 3.17  t o be n - d i m e n s i o n a l .  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 =0,l,2,..., Ix <N, ^ >0,IA <1. i  Here  (2)  ±  B(x,X) =  (1-X)  1  1  N-x  Xx T|)(X,X)  (N-x)!  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:  for  . x = 0 I y 2 ^ . . ; .^x^.S; X ,=I,2/... . , £ A < M . x  s  J  slight-  ±  - 45 -  B(x,x)  Here  Negative  . (JJ-x)  =  '  ^  ( A  '  x )  ( )  =  fixed. M,N:  m u l t i v a r i a t e hypergeometric with N I T ( M ) r(M+N-yx.-yx.)  <f,(x;A)  ~  =  r(N+M) for x =0,l,2,... 1  ,  r(x.+x.)  ~  TT  (N-Jx )ir(M-x ) 1  Ix <N, X > 0 , 1  x !r(x )  1  i  Here B(x,X) = M u l t i v a r i a t e negative r(N+Yx «>(xjA)  1  r (xx* +r (xX) )  , ^(X,x) = binomial with fixed -N-Jx  ) —d+Ix.)  =  r(N)  X  1  =  X  N : i  X  J J — — x : ±  f o r x,=1,2. .. . ,AA.SO . x ' 3. B(x,X)  1  JA <M.  ±  ( TN(-Mx +) !NT- (X M- X- X) )  Here  '  x  r (N+x) (1+X )  i  ~  N  x  ,  x  i|>(X,x)  =  —  x! Inverted D i r i c h l e t with fixed r(e+yx.) d>(x;X)  =  6 :  x  ^  /  r(9)(i+£x.) 2>i  1  r  (  x  i  }  i (.:^0, X0>0 . X  1  -L  1  B(x,X)  Here  Dirichlet  j  t  = T< 9 + " X ) (1+x ) ~  compound n e g a t i v e  >\  <j>(x;X) =  for  -  TT— 6+  for  1  r(N+9) v T' IN  r(N)r(e)  6 - X  ,.  I|J(X,X)  =  *  r(x)  multinomial  with  fixed N ,  rO+Jx. )r(N+Jx.) r(x.+x.) " 1 " 1 -t—r ± _L J | r(N+9+JX +JX ) x :r(x )  x . = 0 , l , 2 , .. . , 1X0>0 . l l  1  ±  1  1  - 46 r (K+x)r (6+A )  Here  B(X,A)  P r o p o s i t i o n 3.22. to  (6) d i s c u s s e d  =  r(N+e+x+x)  U  1 5  ...,A ) 4  x!r(A)  !  1  m  m  m  Pr{p' < (or< )X^< (or< ) r 1  (3)  =  ( ^)(A ,...,A )  m  decreasing  (X ,x)  3  (X-^, . . . ,X | A-|_, . • . ,X )  (2)  ty  I f X h a s i t s d i s t r i b u t i o n i n t h e f o r m o f (1) A a b o v e , t h e n f o r m<n: I  (1)  r (X+A ) ,  st>^ ( s t ^ ) (X-j^, . . . , X | A-j^, . . • , A ) m  m  f o r i = l, . . . ,m| A-^, . . . ,A } i s a  2  Schur concave f u n c t i o n o f (A-^,...,A ). m  P r {number o f z e r o components o f ( X ^ , . . . , X ) > k | A-^, . . . ,A } m  is a decreasing Proof  : (1) f o l l o w s  cerning  the function  same l i n e as t h a t  m  Schur convex f u n c t i o n o f (A-^,...,A ). m  f r o m Theorem 3.13, 3.16 a n d t h e r e m a r k B .  The p r o o f s  o f P r o p o s i t i o n 3-l8,  for  con-  (2) a n d (3) f o l l o w t h e  together  with  the appli-  c a t i o n o f (1). E.  Application to hypothesis t e s t i n g . C o n s i d e r a S c h u r c o n v e x f a m i l y , p a r a m e t r i z e d by A_, A^>0.  G i v e n A- +...+A =nk w h e r e L  n  k  i s either a p o s i t i v e integer or  j u s t a p o s i t i v e number, d e p e n d i n g on t h e f o r m o f A, we want t o t e s t t h e n u l l h y p o t h e s i s H : A =...=A Q  H-^: n o t a l l t h e A^ e q u a l Intuitively, that  1  =k v e r s u s t h e a l t e r n a t i v e  k' .  symmetry o f t h e d i s t r i b u t i o n f u n c t i o n  i f t h e A^ a r e e q u a l ,  implies  t h e n i t i s more p r o b a b l e t h a t t h e  components X^ do n o t d i f f e r g r e a t l y , w h i c h a l s o means  that  -  X. / £x.  do n o t d i f f e r  117  greatly.  -  Formulated I n terms  f u n c t i o n s , t h i s means t h a t when  P  o f Schur  i s Schur convex  (Schur  c o n c a v e ) , F ( X ^ / £ x . ,. . . , X / J X j . ) t e n d s t o be s m a l l e s t  (largest)  n  under H Q .  So we c a n t a k e a s a t e s t  function  0 where  F  i s some S c h u r c o n v e x  0<y<l a n d is  <(>)  <  c  (Schur concave)  function,  a r e chosen so as t o a c h i e v e t h e d e s i r e d  easy t o v e r i f y  that  T  i s a Schur convex  size.  It  f u n c t i o n i n both  c a s e s , s o t h a t t h e power f u n c t i o n 3 (A_) = E ^ T ( X ) i s S c h u r T  convex point  Therefore, 3  i n A.  m  (k,...,k). ' Consequently, the test i sunbiased. The  case o f t h e m u l t i n o m i a l  F(x /^Xj,...»x /£xj) 1  is  a t t a i n s i t s minimum a t t h e n u l l  first  n  with  - P(x /N,...,x /N) = [ h ( x ) , 1  n  1  f o u n d by Cohen a n d S a c k r o w i t z (1975).  t i o n o f £h(x^) t o S c h u r c o n v e x Perlman and R i n o t t also point ratio  family,  (to appear).  functions  h  convex,  A generaliza-  G ( x ) i s f o u n d by  Cohen a n d S a c k r o w i t z (1975)  out that t h e c l a s s Jh(x^)  includes  the likelihood  t e s t w i t h h ( x ) = x l l o g x and t h e goodness o f f i t t e s t w i t h 2 h ( x ) = x . We c a n v e r i f y t h a t t h e g o o d n e s s o f f i t t e s t w i t h 2 h{x)-x also applies to the hypergeometric d i s t r i b u t i o n .  - 48 -  Bibliography  [1]  C h o n g , K.M. (1976). An i n d u c t i o n t h e o r e m f o r r e a r r a n g e ments. C a n a d i a n J o u r n a l o f M a t h e m a t i c s 28, 154-160.  [2]  C o h e n , A. a n d S a c k r o w i t z , H.B. (1975). U n b i a s e d n e s s 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 goodn e s s o f f i t t e s t s f o r t h e e q u a l c e l l case'. Annals o f S t a t i s t i c s 3, 959-964.  [3]  F u c h s , L. (1947). A new p r o o f o f a n i n e q u a l i t y o f Hardy-Littlewood-Polya. • Matematisk T i d s s k r i f t 53-54.  B,  [4]  H a r d y , 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 c o n v e x functions. M e s s e n g e r o f M a t h e m a t i c s .,58 ,- -145-152.  [5]  H a r d y , G.H., L i t t l e w o o d , J . E . a n d 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 University Press.  [6]  H o l l a n d e r , M. , P r o s c h a n , F.. a n d S e t h u r a m a n , J . ( t o ap-pear). 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 ranking problems.  [7]  M a r s h a l l , A.W. a n d O l k i n , I . ( t o a p p e a r ) . via majorization.  [8]  M a r s h a l l , A.W. a n d 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 o f M a t h e m a t i c a l A n a l y s i s a n d A p p l i c a t i o n s 12, 87-90.  [93  M i r s k y , L. (1959). Inequalities f o rc e r t a i n class of convex f u n c t i o n s . P r o c e e d i n g s o f E d i n b u r g h Mathem a t i c a l S o c i e t y 11, 231-235.  Inequalities  [10]  M i t r o n o v i c , D.S. (1970). A n a l y t i c S p r i n g e r - V e r l a g , New Y o r k .  inequalities.  [11]  M u d h o l k a r , G.S. (1966). The i n t e g r a l o f a n 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 o f A m e r i c a n 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. a n d S e t h u r a m a n , 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 majorization. A n n a l s o f S t a t i s t i c s 5, 263-273. A l s o , F l o r i d a State U n i v e r s i t y Technical Report M306, (1974).  [13]  N e v i u s , S.E., P r o s c h a n , F. a n d S e t h u r a m a n , 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 R e p o r t 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 o f D i r i c h l e t Integrals with a p p l i c a t i o n s to the multinomial d i s t r i b u t i o n and t h e a n a l y s i s o f 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). S u r q u e l q u e s a p p l i c a t i o n s des f u n c t i o n s c o n v e x e s e t c o n c a v e s au s e n s de I . Schur. J o u r n a l de M a t h e m a t i q u e s P u r e e t 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 a p p e a r ) On t h e u n biasedness of goodness-of-fit t e s t s .  [17]  P r o s c h a n , F. and S e t h u r a m a n , 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 t h e o r e m . Florida S t a t e U n i v e r s i t y T e c h n i c a l R e p o r t 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 r e a r r a n g e m e n t 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 Journal of M a t h e m a t i c s 15, 60-77.  [19]  T o m i c , M. (1949). G a u s s ' t h e o r e m on t h e c e n t r o i d a n d 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 d e s M a t h e m a t i c i e n s e t 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 Y u e , P.C. (1973). A m a j o r i z a t i o n t h e o r e m f o r t h e number o f d i s t i n c t o u t c o m e s 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 M a t h e m a t i c s 6_, 391-398.  

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