Guttman I. a Bayesian Analogue of Paulsons Lemma and Its Use in Tolerance Region Construction When Sampling From the Multi-Variate Normal 1971

8/12/2019 Guttman I. a Bayesian Analogue of Paulsons Lemma and Its Use in Tolerance Region Construction When Sampling

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A BAYE SIAN ANA LOG UE OF PAULSON S LEMMA A ND ITS USEIN TOLERANCE REGION CONS TRUCTION WHEN SAM PLING

FROM THE MULTI-VARIATE NORMAL

IRWIN G UTTMAN(Re ceiv ed Feb . 16, 1970)

1 . I n t r o d u c t i o nS u p p o s e w e a r e s a m p l i n g o n a k d i m e n s i o n a l r a n d o m v a r i a b l e X ,

d e f i n ed o v e r R ~, E u c l i d e a n s p a c e o f k d i m e n s i o n s , a n d l e t { A } - - ~ d e -n o t e a a - a l g eb r a o f s u b s e ts A o f R ~. W e a s s u m e t h a t ~ I c B - - B o r e ls u b s e t s o f R ~.

S u p po s e f u r t h e r t h a t X h a s t h e a b s o l ut e l y co n t i n u o u s d i s t r ib u t i o n( I . I ) F ( x [ @ ) = I ' ~ - ' I I f ( Y l , ' , y ~ 1 6 ) d y , . . . d y ,a n d # ~ I2, w i t h 9 a n i n d e x i n g s e t w h i c h w e w i ll r e f e r t o a s t h e p a r a m -e t e r sp a ce . W e d e n o t e a r a n d o m s a m p l e o f n i n d e p e n d e n t o b s e r v a ti o n so n X b y ( X ~ , . . . , X ~) o r { X ~ } . W e h a v e t h e f o l l o w i n g d e f i n it i o n s .

D E FIN IT IO N 1 .1 . A s t a t i s t i c a l t o l e r a n c e r e g i o n S ( X ~ , . . . , X , ) i s as t a t i s t i c d e f i n e d o v e r R ~ . . . w h i c h t a k e s v a l u e s i n t h e a -a l g e b r a ~ .

T h i s d e f i n i t io n , t h e n , i m p l i e s t h a t a s t a t i s t i c a l t o l e r a n c e r e g i o n i s as t a ti s ti c w h i c h i s a s e t f u n c t i o n , a n d m a p s " t h e p o i n t " ( X 1 , - . . , X ~)R ~ i n t o t h e r e g i o n S ( X ~ , . . . , X ~ ) ~ ~ , t h a t i s, S ( { X ~ } ) c R ~. W h e n c o n -s t r u c t i n g s u c h st a t i s ti c a l t o l e r a n c e re g i o n s , v a r i o u s c r i t e r i a m a y b e u s e d .O n e t h a t i s o f t e n b o r n e i n m i n d i s c o n t a i n e d i n t h e f o l l o w i n g d e fi n i ti o n .

S ( X 1 ,- 9 X , ) i s a ~ - e x p e c t a t i o n s t a t i s t ic a l t o l e r a n c eEFINITION 1.2 .r e g i o n i f1 . 2 )

f o r a l l 0 ~ Q , w h e r eE i X ,} { F [ S ( X 1 , , X ,) [ 01} = fl

( 1 .2 a ) F [ S [ @ ] = f ' ~ " f f ( u l 6 ) d u .* This research was supported in part by the Wisconsin Alum ni Research Foundation.Present address is: Centre de Recherches Math~matiques, Universit~ de M ontreal.

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8 IRWIN GUTT MAN

T h e q u a n t i t y F [ S ] O ] = F [ S ( X I , . . . , X , ) I O ] i s c a l l e d t h e c o v e r a g e o ft h e r e g i o n S , a n d w i ll b e d e n o t e d b y C[S] . We n o t e t h a t i t c an b ev i e w e d a s t h e p r o b a b i l i ty o f a n o b s e r v a t i o n , s a y , Y , f a l li n g in S , w h e r eY i s i n d e p e n d e n t o f X ~ , . . . , X , a n d , o f c o u r se , h ~ s d is t r ib u t i o n F ( y [ 0 ) .N o w b e c a u s e S ( X 1 , . . - , X , ) i s a r a n d o m s e t f u n c t i o n , F [ S ( X ~ , . . . , X , ) I 0 ]i s a r a n d o m v a r i a b l e a n d h a s t h e d i s t r i b u t i o n o f it s o w n . H e n c e , c o n -s t r u c t i n g a n S t o s a t i s f y ( 1 .2 ), s i m p l y im p l i e s t h a t w e a r e i m p o s i n g t h ec o n d i ti o n t h a t S b e s u c h t h a t t h e d i s t r i b u t io n o f i ts c o v e r a g e F[S]O]h a s e x p e c t a t i o n ( m e a n v a l u e ) {3. P a u l s o n [ 6] h a s g i v e n a v e r y i n t e r e s t -i n g c o n n e c t i o n b e t w e e n s t a t i s t ic a l t o l e r a n c e r e g i o n s a n d p r e d i c t i o n r e g io n s .

PAULSON S LEMMA. I f On the basis of a given sample on a k-dimen-sion~d ran dom variable X, a k-dimension al "conf idence" region S(X I ,9 . , X , ) o f l evel ~ i s fo un d for s ta t is t i c s T I , . . . , T~ , where T ~ - T~(Y1,9 . , Yq) , where the ( k vector observations Y~, j = l , . . . , q are independ-ont observa tions on X , and independent o f X I , . . . , X , , and i f C i s de-f ined to be such thatC= I s dG(t)1.3)

where G( t ) i s t he d i s t r ibu t ion func t ion o f T ( T = ( T 1 , . - . , T ~) , a n d T ~=T , Y , , . . . , t h e n1 . 4 ) .

B e f o r e w e p r o v e t h i s l e m m a , w e r e m a r k t h a t o u r i n t e r e s t w i l l b ef o r t h e c a s e q = l , ( t h a t i s, w e w i l l h a v e o n e f u t u r e o b s e r v a t i o n Y~ = Y ) ,an d T~ = T ~(Y ) = Y~ so t h a t T = Y an d G ( t ) = F ( y [ 0 ). H e n c e , C g i v e nb y ( 1. 3) i s s i m p l y t h e c o v e r a g e o f S ( X I , . . . , X , ) . N o t e t h a t C d e p e n d sh e r e o n S a n d 0 a n d i n d ee d w e m a y w r i t e C= C, ( S ) .

I n t h e s e c i r c u m s t a n c e s , t h e n , P a u l s o n s L e m m a t h e n g i v e s u s a no p e r a t i o n a l m e t h o d f o r c o n s t r u c t i n g a s t a t i s t i c a l t o l e r a n c e r e g i o n o f ~ -e x p e c t a t i o n , n a m e l y :

F i n d S , a p r e d i c t i o n r e g i o n o f l e v e l ~ f o r a f u t u r e o b s e r v a t i o n Y .I f t h i s i s d o n e , t h e n S i s a t o l e r a n c e r e g i o n o f ~ - e x p e c t a t i o n .

P R OO F O F P A UL S ON S L E M M A . T h e j o i n t d i s t r i b u t i o n f u n c t i o n o fX ~, . . . , X , , i sn r x , l e )

N o w t h e l e f t - h a n d s i de o f (1 .4 ) m a y b e w r i t t e n a s(1 .5) E[C ]= f s ~ i s dG ( t)d ~ F( x , , e ) .


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A BAYESIAN ANALOGUE OF PAULSON S LEMMA AND ITS USE 6gB u t t h e r i g h t - h a n d s i d e o f ( 1.5 ) i s t h e p r o b a b i l i t y t h a t T l ie s i n S , a n dw e a r e g i v e n t h a t S = S ( X ~ , . . . , X , ) i s ~ - l eve l con f idence r eg ion fo r T( o r p r e d i c t i o n r e g i o n f o r T ) . H e n c e , t h e r i g h t - h a n d s i d e o f (1 .5 ) h a sv a l u e ~ , a n d w e h a v e t h a t E [ C] = ~ .

A s a n i l l u s t r a t i o n o f t h e a b o v e , w e t a k e t h e c a s e q = l , a n d s u p p o s et h a t s a m p l i n g i s o n t h e k - d i m e n s i o n a l n o r m a l v a r i a b l e N ( ~ , 2 ). I t i sw e l l k n o w n t h a t a 1 0 0 ~ % p r e d i c ti o n r e g io n f o r Y , w h e r e Y = N ( p , X),c o n s t r u c t e d o n t h e b a s i s o f t h e r a n d o m s a m p l e o f n i n d e p e n d e n t o b s e r -v a t i o n s X I , - . . , X , [ w h e r e Y, X ~ , . . . , X , a r e a l l i n d e p e n d e n t ] i s1 . 6 ) S( [X ,} ) = { Y[ ( Y - . X ) V - ( Y - .X ) ~ _C ~ }

w h e r e t h e m e a n v e c t o r 2 ~ a n d t h e s a m p l e v a r i a n c e - c o v a r ia n c e m a t r i x Va r e d e f i n e d b y

1 . 6 a )9

r = ( n - 1 ) -w i t h C ~ g i v e n b y(1 .6b ) C~ = [ ( n - 1)k/(n - k)] [1 + n-L]Fk,,,_k;~-~a n d F ~ ,,_ ~:I_ ~ i s t h e p o i n t e x c e e d e d w i t h p r o b a b i l i ty 1 - ~ w h e n u s i n gt h e S n e d e c o r - F d i s tr i b u t io n w i t h ( k, n - - k ) d e g r e e s o f f r e e d o m . H e n c e ,b y P a u l s o n ' s L e m m a , S g i v e n b y ( 1 .6 ) is a E - e x p e c t a t i o n t o l e r a n c e r e -g i o n. T h i s r e g i o n is k n o w n to h a v e c e r t a i n o p t i m u m p r o p e r t i e s - - s e e ,f o r e x a m p l e , F r a s e r a n d G u t t m a n [1 ].

W e n o w a p p r o ac h t h e p r o b l e m o f c o n s t r u c t i n g ~ - e x p e c t a t i o n t o l e r-a n c e r e g i o n s f r o m t h e B a y e s i a n p o i n t o f v i e w . W e w i ll s e e t h a t t h e r ei s a d i r e c t a n a l o g u e o f P a u l s o n ' s L e m m a , w h i c h a r i s e s i n a n a t u r a l a n di n t e r e s t i n g w a y .

2 The Bay esian approachI n th e B a y e si a n f r a m e w o r k , r e f e r e n c e s a b o u t t h e p a r a m e t e r s @ i n

a s t a t is t ic a l m o d e l a r e s u m m a r i z e d b y t h e p o s t e r i o r d i s t r i b u t io n o f t h ep a r a m e t e r s w h i c h a r e o b ta i n e d b y t h e u s e o f a t h e o r e m d u e t o t h eR e v e r e n d T h o m a s B a y e s. T h i s t h e o r e m , a s im p l e s t a t e m e n t o f c o n d i-t i o n a l p r o b a b i l it y , s t a t e s t h a t t h e d i s t r ib u t i o n o f t h e p a r a m e t e r s 0 , g i v e nt h a t X ~ i s o b s e r v e d t o b e x~ , i - - 1 , - . . , n , i s(2.1) p [ o l x , } = c p e ) p [ I x , } l e ] ,w h e r e t h e n o r m a l i z i n g c o n s t a n t c i s s u c h t h a t


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7 IRWIN GUTTMAN

(2 .1a) c - = l o p o ) p [ { x , } I O ] d O ,p ( 0 ) i s t h e ( m a r g i n a l ) d i s t ri b u t io n o f t h e v e c t o r o f p a r a m e t e r s 0 , a n dp [ { x d [ 0 ] is t h e d i s t ri b u t i o n o f t h e o b s e r v a t io n s { X ,} , g i v e n 0. W h e nw e a r e i n d e e d g i v e n t h a t { X ~ } = { x~ }, t h e n w e o f t e n c a ll p [ { x d [ 0 ] t h el i k e li h o o d f u n c t i o n o f 0 a n d d e n o t e i t b e l [ 0 [ { x d ] .

T h e i n g r e d i e n t s o f ( 2. 1) m a y b e i n t e r p r e t e d a s f o ll o w s . T h e d i s t r i -b u t i o n p (O ) r e p r e s e n t s o u r k n o w l e d g e a b o u t t h e p a r a m e t e r s 0 b e f o r e t h ed a t a a r e d r a w n , w h i l e l[ 0 [ { x d ] r e p r e s e n t s i n f o r m a t i o n g i v e n t o u s a b o u t0 f r o m t h e d a t a { x ~} , a n d f i n al ly , p [ O [ { x d ] r e p r e s e n t s o u r k n o w l e d g eo f 0 a f t e r w e o b s e r v e t h e d a t a . F o r t h e s e r e a s o n s, p ( O ) i s c o m m o n l yc a l l e d t h e a - p r i o r i o r p r i o r d i s t r i b u t i o n o f 0 , a n d p [ O l { x d ] , t h e a -p o s t e r i o r i o r p o s t e r i o r d i s t r ib u t i o n o f 0 . U s i n g t h is i n t e r p r e t a t i o n , t h e n ,B a y e s t h e o r e m p r o v i d es a f o r m a l m e c h a n i s m b y w h i c h o u r a -p r i o r i i n-f o r m a t i o n i s c o m b i n e d w i t h s a m p l e i n f o r m a t i o n t o g i v e u s t h e p o s t e r i o rd i s t r i b u t i o n o f 0 , w h i c h e f f e ct i v e ly s u m m a r i z e s a ll t h e i n f o r m a t i o n w eh a v e a b o u t 0 .N o w t h e r e a d e r w i l l r e c a l l t h a t S i s a t o l e r a n c e r e g i o n o f f i -e x p e c t a-t i o n i f i ts c o v e r a g e C [ S ] h a s e x p e c t a t i o n ~ . N o w , f r o m a B a y e s i a n p o i n to f v i e w , o n c e h a v i n g s e e n t h e d a t a , t h a t i s , h a v i n g o b s e r v e d {X ~} = { x d ,t h e n C [ S ] = I f ( y [ O ) d y i s a f u n c t i o n o n l y o f t h e p a r a m e t e r s 0 , a n d t h eJ 3e x p e c t a t i o n r e f e r r e d t o i s t h e e x p e c t a t i o n w i t h r e s p e c t t o t h e p o s t e r i o rd i s t r i b u t i o n o f t h e p a r a m e t e r s 0 , t h a t i s , on t h e b a s is o f t h e g i v e n d a t a{ x~ }, w e w i s h t o c o n s t r u c t S s u c h t h a t(2.2) E [ C [ S ] i { x } l = fo I s f ( Y l O ) P [S l { x ] l d Y g O = ~w h e r e Y h a s t h e s am e d i s tr i b ut i on a s t h e X , , n a m e l y f ( - [ 0 ) . N o w i ti s i n t e r e s t i n g t o n o t e t h a t , a s s u m i n g t h e c o n d i ti o n s o f F u b i n i s T h e o r e mh o ld , s o t h a t w e m a y i n v e r t t h e o r d e r o f i n t e g ra t io n , w e h a v e t h a t

E [ C [ S ] ] {x,]] = f s Io f ( y [ O)p[O [ {x ,] ]40 @2.3)= Is h [ y i { x , } ] g y .

N o w t h e d e n s i t y h [ y [ [ x ~ ] ] , w h e r e(2.4) h [ y l {x,} ] = IQ f ( y I @ )p [O ] {x,} ]40i s ( e x a m i n i n g t h e r i g h t - h a n d s i d e o f ( 2 . 4 ) ) s i m p l y t h e c o n d i t i o n a l d i s t r i -b u t i o n o f Y , g i v e n t h e d a t a [ x ~ }, w h e r e Y m a y b e r e g a r d e d a s a n a d -d i t i o n a l o b s e r v a t i o n f r o m f ( x [ O ) , a d d i t i o n a l t o a n d i n d e p e n d e n t o f X 1 ,


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A BAYESIAN ANALO GUE OF PAUL.SON S LEMM A AND ITS USE 71

9 . , X , . T h i s d e n s i t y h[yl{x }] i s t h e B a y e s i a n e s t i m a t e o f t h e d i s t r i -b u t i o n o f Y a n d h a s b e e n c a ll e d t h e p r e d i c t i v e o r f u t u r e d i s t r i b u t i o n o f Y .( F o r f u r t h e r d i sc u ss io n , s e e , f o r e x a m p l e , G u t t m a n [ 4] a n d t h e r e f e r e n c e sc i t e d t h e r e i n . )H e n c e , ( 2 . 2 ) a n d ( 2 . 3 ) h a v e t h e v e r y i n t e r e s t i n g i m p l i c a t i o n t h a t Si s a f l - expec t a t ion t o l e r ance r eg i on i f i t i s a ( p r ed i c t i ve ) ~ -con f i dence r e -g i o n f o r Y , w h e r e Y h a s t h e p r e d i c t i v e d i s t r i b u t i o n h [ y l { x , } ] de f i nedb y ( 2. 4) . T h i s is t h e B a y e s i a n a n a l o g u e o f P a u l s o n ' s L e m m a g i v e n i nS e c ti o n 1 w i t h q = l . T o r e p e a t in a n o t h e r w a y , f o r a p a r ti c u l a r f , w en e e d o n l y f i n d h [ y l { x , } ] a n d a r e g i o n S t h a t i s s u c h t h a t(2.5) Pr Y e S ) = I s h [ y [ { x, } ] d y - f l .W e s u m m a r i z e t h e a b o v e re s u l ts i n t h e f o l l o w i n g l e m m a .

LEMMA 2.1. I f on the bas is o f observed da ta {x~} , a predic t ive ~-confidence reg ion S = S {x~ }) i s cons truc ted such tha t(2.6) Is h[y l {x~} dy-=flwh e r e th e p r e d ic t iv e d i s t r ib u t io n i s g iv e n b y (2.4), a n d i f C [ S ] i s t h ecoverage o f S , tha t i s(2.7) C [ S ] = C [ S x , , - . . , x~ ) ] = I s f y [O)dywh e r e f i s th e c o mm o n d i s t r ib u t io n o f th e in d e p e n d e n t r a n d o m v a r ia b le s,X ~ , . . . , X~ , Y , th e n th e p o s te r io r e x p e c ta t io n o f C[ S ] i s ~ , th a t i s S i sof ~-expectation.

(The proof i s s imple and u t i l i zes re la t ions (2 .3) and (2 .6) . )

3. Sampl ing f ro m the k var ia te normalW e s u p p o s e i n t h is s e c t i o n t h a t s a m p l i n g is f r o m t h e k - v a r i a t e n o r -m a l N ( p , Z ) , w h o s e d i s tr i b u t i o n i s g i v e n b y

3 . 1 ) f x I P , 2 ) = ( 2 ~ ) - ~ n ] $ - I 1 n e x p f - 1 ( x - / J ) ' ~ : - l ( x - - p ) } ,w h e r e / J is ( k x 1 ) a n d ~7 i s a ( k x k ) s y m m e t r i c p o s i ti v e d e f i n it e m a t r i x .I t i s c o n v e n i e n t t o w o r k w i t h t h e s e t o f p a r a m e t e r s (/~ , 2 -1 ) h e r e , a n da c c o r d i n g l y , s u p p o s e t h a t t h e p r i o r f o r t h i s s i t u a t i o n i s t h e c o n j u g a t ep r i o r o r R a i f f a a n d S c h l a i f e r g i v e n b y


6/10

7 2 I R W IN G U T T M A N

( 3 .2 ) P ( P , X - ' ) d ~ -~ ~ I ~ - ' I9 e x p { - - 8 9 r I : - ~ [ ( n o - 1 ) V o + n o ( p - ~ ) ( p - Y c o ) ' ] I d p d - ~

w h e r e x0 i s a ( k x 1 ) v e c t o r o f k n o w n c o n s t a n t s a n d 110 i s a ( k s y m -m e t r i c p o si ti v e d e f in i te m a t r i x o f k n o w n c o n s ta n t s * . I t i s t o b e n o t e dt h a t i f ~ t e n d s t o 0 a n d 0 h - l ) V 0 t e n d s t o th e z e ro m a t r ix , t h e n ( 3 .2 )t e n d s t o t h e " i n - i g n o r a n c e " p r i o r a d v o c a t e d b y G e i ss e r [ 2 ] a n d G e i s se rand Corn f i e ld [3 ] , v i z(3 .3) p(p , $-~ )dpd ~ -~ c~ [$-~ [ - ( ~ + , / 2 d p d ~ r ' ~ .N o w i t is e a s y t o s e e t h a t i f n i n d e p e n d e n t o b s e r v a t io n s X ~ a r e t a k e nf r o m ( 3 .1 ) , a n d w e o b s e r v e {X ~} t o b e [ x~ ] , t h a t t h e l i k e li h o o d f u n c t i o ni s g i v e n b y(3 .4 ) l [ p , .~ ' - ' l { x , }l = ( 2 x ) - W Z l ~ - ~ [ ' '

. e x p { - l t r Y ; - '[ ( n - 1 ) V w h e r e

a n dnx = n - l ~ X~

/ . f f i

nn - 1 ) v = ~ . ( x , - ~ ) ( x , - ~ ) .( t h e a b b r e v i a ti o n " t r A " s ta n d s f o r th e t r a c e o f t h e m a t r i x A . ) N o wc o m b i n i n g ( 8 . 2 ) a n d (3 .4 ) u s i n g B a y e s ' T h e o r e m g i v e s u s t h a t t h e p o s -t e r i o r o f ( p , Z - I ) i s s u c h t h a t( 3 . 5 ) p [ ~ , : ~ -' 1 { x , ] ] r 16 2~ -1 p + ~ - ~ - / '

19 ex p { ~ t r 2:-'[(n0 -- 1)1/'0+ ( n - - 1 )V

+ m ( ~ - ~ 0 ) ( ~ - ~ 0 ) + n ( ~ - ~ ) ( ~ - ~ ) ] l oN o w t h e e x p o n e n t o f (3 .5 ) m a y b e w r i t t e n i n s im p l if ie d f o r m o n " c o m -p l e t in g t h e s q u a r e " i n p , t h a t i s, t h e t e r m i n s q u a r e b r a c k e t s i n t h ee x p o n e n t o f ( 3 . 5 ) m a y b e w r i t t e n , a f t e r s o m e a l g e b r a , a s( 3 . 6 ) ( n + n o ) ( ~ - ~ ) (~ - ~ ) ' + ( no - 1 ) F 0 + ( n - 1 ) V + R

* W e a r e i n ef f e ct s a y i n g t h a t o u r p r i o r i n f o r m a t i o n o n p a n d ~ P-I i s s u c h t h a t w ee x p e c t p t o b e ~ 0 , w i t h d i s pe r s io n , t h a t is , v a r i a n c e c o - v a r i a n c e m a t r i x o f p , t o b e ( ~ - - l ) V 0 /[ n o ( n o - - k - - 2 ) ] , a n d t h a t w e e x p e c t ~ - 1 t o b e V o 1 e t c .


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A BAYESIAN ANALOGUE OF PAULSON S LEMMA AND ITS USE 73

w h e r e( 3 . 7 )a n d(3 .7a)

~ = (n -F no) -l ( n~ o + nYc)

R = n ~Yd + noYC o~- (n + no) (&Yd)= n n on + n oH e n c e , w e m a y n o w w r i t e ( 3 . 5 ) a s f o l l o w s :3 . 8 )

T o ) T o ) .

9 e x p I - l [ t r E - ~ Q + ( n o + n ) ( l ~ - ~ ) 'X - ~ ( I J - ~ ) ] }w h e r e(3 .8a) Q = ( n o - 1 ) V o + ( n - 1 ) V + n o n ( x - ~ 0 ) ( x - ~ 0 ) ' ,n o + na n d c i s t h e n o r m a l i z i n g c o n s t a n t n e c e s s a r y to m a k e (3 .8 ) a d e n s i t y , t h a ti s, i n t e g r a t e t o 1 . N o w t o d e t e r m i n e c, w e f i rs t i n t e g r a t e w i t h r e s p e c tt o / J a n d t h e n 2 - I, a n d i n so d o i n g , w e m a k e u s e o f t h e i d e n t i t i e s d e -r i v e d f r o m t h e k - v a r i a t e n o r m a l a n d k - o r d e r W i s h a r t d i s t r i b u t i o n s , v i z

= 2'n~n~c~ -~'/' I M I n ]-[ F [ ( m + 1- - / ) /21 9A s i s e a s i l y v e r i f i e d , p e r f o r m i n g t h e i n t e g r a t i o n y i e l d s

/(3.10) c = ( n + m ) ~ l ' [ Q [


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(3.13)w i t h

74 IRWIN GUTTM AN3.12) W = ( n + m ) ( p - Y c ) ( p - Y c ) ' + ( p - y ) ( p - y ) ' .

A g a i n " c o m p l e t i n g t h e s q u a r e i n p i n (3 .1 2 ), w e e as il y , b u t t e d i o u s l y ,f i n d t h a tW = ( n + n o + l ) ( p - ~ ) ( p - ~ ) ' - } n + n o ( y- Yc )( y- Yc ) 'n + n 0 + l

=(n+no+i)-'[(n+m)i+y].The i n t e g r a t i on w i t h r e sp ec t t o p i n ( 3. 11) , u s i ng (3 .9a) , g i ve s u s

( 3 .1 4 ) [ y l { x d ] = I . . . ( n + m+ l ) - / I I- 1 1 (~ + = o- /n + ~ + l

I n t e g r a t i n g ( 3 . 1 4 ) w i t h t h e h e l p o f ( 3 . 9b ) , a n d s u b s t i t u t i n g f o r t h e v a l u eo f c g i v e n b y ( 3. 1 0 ) , w e f i n d t h a t3 . 1 5 ) h [ y l { x , } ] = ( n + m ) ' ; a F t ( n + m ) / 2 l l Q - * l ~n no 1 II ' / ' F[ (n no -- k)/2]

9 I t n + no Q_ ~(y_Y c)(y_Yc) , l -'"+~o'nn + n 0 N o w u s i n g t h e i d e n t i t y ( p r o v e d i n t h e a p p e n d i x )(3.16) I I ~ I- -A B [ = I , , - B Aw h e r e A i s (n~ a n d B i s ( n , n ~ ), w e h a v e t h e r e s u l t t h a t t h e p r e -d i c t i v e d e n s i t y o f Y i s g i v e n b y8 . 1 7 ) h [ y l i x , } l = ( n + n o ) r [ ( n + m ) / 2 l l Q - ' l ' / 'n l I I ' / 2F[ ( n

9 n + n 0n + n 0 + lt h a t i s, w e h a v e t h e i n t e r e s t i n g re s u l t t h a t t h e p r e d ic t iv e d i s t ri b u t i o no f Y , g i v e n { x~ }, i s r e l a t e d t o t h e k - v a r i a t e t - d is t ri b u t io n , d e g r e e s o ff r e e d o m ( n + n o - k ) . A s m a y b e s e e n f r o m p r o p e r t i e s o f t h e m u l t i v a r i -a t e - t ( s e e , f o r e x a m p l e , T i a o a n d G u t t m a n [ 8 ] ) , w e h a v e t h a t

b F~ . . + ~_~o + n ( Y -Y c . ) ' Q - '( Y - I c )= n + n o - k(3.18) n + n 0 + iS u p p o s e n o w t h a t w e a r e i n t e r e s t e d i n t h e c e n t r a l 1 00 ;9 o f t h e

n o r m a l d i s t r i b u t i o n ( 3 .1 ) , t h a t is , i n t h e s e t


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A BAYESIAN ANALOGUE OF PAULSON S LEMMA AND ITS USE 75

(3.19) A ~ = { y l ( y - - p ) 2 - ~ ( y - - p ) < =Z ~,~_~ }w h e r e Z ~,l_p i s t h e p o i n t e x c e e d e d w i t h p r o b a b i l i ty ( 1 - 8 ) w h e n u s i n gt h e c h i - s q u a r e d i s tr i b u t i o n w i t h k d e g r e e s o f f r e e d o m . G i v e n p a n d 2o r (~ , 2 -~ ), w e h a v e t h a t(3.20) P Y e A~lp, 2 =~t h a t i s, i f w e k n e w ( ~ , 2 ) , A k w o u l d b e a 1 0 0 ~ % p r e d i c t i v e r e g i o n f o rY . N o w s in c e w e d o n ' t k n o w ( ~ , 2 ) , w e u s e a s a n e s t i m a t o r o f t h eden s i t y ( 3 .1 ) , t he de ns i t y (3 .17 ), a f t e r ob se r v i ng t he da t a { x~} . H ence ,a s ens i b l e p r ed i c t i ve r eg i on f o r Y i s t he c en t r a l e l li p so ida l r eg i on o f( 3 . 17 ) , nam e l y t he r eg i on(3.21) S ( x , , . . . , x ~ ) _ _ _ t y ] n o . - l - n ( y _ Y c ) ,[ Q / ( n o _ l _ n _ k ) l -l ( y _ ] c )n o - f - n - t - 1

and i t i s e a sy t o s ee , f r om ( 3 . 18 ) , t ha t( 3. 22 ) P ( Y e S I { x~ } ) = ~ .Thus , by Le m m a 2 .1 , w e ha ve t h a t S de f i ned by ( 3 .21) i s a t o l e r an cer eg i on o f ( pos t e r i o r ) ~ - expec t a t i on .W e n o t e t h a t i f n o = O a n d ( n 0 - 1) V 0 i s t h e z e r o m a t r i x , t h a t i s, i f t h es o -c a ll ed i n - i g n o r a n c e p r i o r g i v e n b y ( 3 .3 ) i s t h e a p p r o p r i a t e p r i o r, t h e nt h e a b o v e r e s u l t s im p l y t h a t t h e ~ - e x p e c t a t i o n r e g i o n i s o f t h e f o r m(3.23) S ( { x , } ) - - I y l n - ~ ( y - Y c ) [ ( n - 1 ) V / ( n - k ) ] - ( l t- Y c ) ~ _ k F , ,, , _ , : , _ , tw h i c h i s i n t e r e s t i n g , s i nc e t h i s l a t t e r r e s u l t i s i n a g r e e m e n t w i t h t h esam pl i ng t heo r y r e su l t (1 .6 ), a s m ay be ea s i l y ve r if i ed .

I t i s t o b e f in a ll y r e m a r k e d , t h a t t h e l e m m a o f S e c t i o n 2 i s q u i t eg e n e r a l a n d m a y b e u s e d w h e n s a m p l i n g is f r o m a n y p op u l at io n . I nf a c t, t h e c a s e o f t h e s i n g l e e x p o n e n t i a l i s d is c u s se d i n G u t t m a n [5 ].

ppendixW e g i ve a p r oo f , due t o G e or ge T i ao , o f ( 3.16 ). Con s i de r t he m a t r i x

eq ua t io ns (A is (n~ x n~) an d B is (n~ x n~))


10/10

7 6a n d

I R W IN G U T T M A N

< A .2 ) B I . j L - B z . j z , jNo~v tak in g de te rm inants o f bo th s ides of A .1) and A .2) y ie lds

I M I . I X ~I1 .1 X ~ I = 1 X . I - I I ~ - B A 1A.3) a nd

whereM [ I~I A

Us ing A .3) we have the r esu l tI ~ - A B I - - I ~ - B A I = I M I .

UNIVERSITY OF WISCON SIN AND UNIVERSITY OF MASSACHUSETTS

R E F E R E N C E S[ 1 ] F r a s e r , D . A . S . ~ n d G u t t m a n , I r w i n 1 95 6). T o l e r a n c e r e g i o n s , Ann . Math . S ta t i s t .27 , 162-179 .[ 2 ] G e i s s e r , S . 1 9 65 ). B a y e s i a n e s t i m a t i o n i n m u l t i v a r i a t e a n a l y s i s , Ann . Math . S ta t i s t .3 6 150-159.[ 3 ] G e i s s e r, S . a n d C o r n f i e l d , J . 1 9 6 3 ) . P o s t e r i o r d i s t r i b u t i o n s f o r m u l t i v a r i a t e n o r m a lp a r a m e t e r s , ]our. Roy. Statist . S o c . , S e r . B , 2 5 , 3 6 8 - 3 7 6 .[ 4 ] G u t t m a n , I r w i n 1 96 7) . T h e u s e o f t h e c o n c e p t o f a f u t u r e o b s e r v a t i o n i n g o o d n e s s - o f -f i t p r o b l e m s , four. Roy . S tat i s t . Soc . Se t . B 2 9 , 8 3 - 1 0 0 .[ 5 ] G u t t m a n , I r w i n 1 96 8) . T o l e r a n c e r e g i o n s : A s u r v e y o f i t s l i t e r a t u r e . V I . T h e

B a y e s i a n a p p r o a c h , Te.ch. Rzp. N o . 1 2 6 , / ~ p a ~ n ~ t o f Statist ics University of Wisconsin.[ 6 ] P a u l s o n , E . 1 9 4 3 ). A n o t e o n t o l e r a n c e l i m i t s , Ann . Math . S ta t i s t . 1 4 , 9 0 - 9 3 .[ 7 ] R a i f f a , H . a n d S c h l a i f e r , R . 1 9 61 ). Ap plie d Statist ical Decision Theory H a r v a r d U n i -v e r s i t y P r e s s .[ 8 ] T i a o , G . C . a n d G u t t m a n , I r w i n 1 96 5). T h e m u l t i v a r i a t e i n v e r t e d b e t a d i s t r i b u t i o nw i t h a p p l i c a t i o n s , Jour. Am er . S tati s t. Assoc . 60 7 9 3 - 8 0 5 .

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