Reference Priors in a Two-Way Mixed-Effects Analysis of Variance Model

2 002 , V ol. 13, N o.2 p p . 31 7～328

R e f e re n c e P rior s in a T w o - W ay M ix e d - E ff e c t s A n aly s i s o f V ari an c e M o de l

In H on g Ch an g ¹⁾ an d B y un g H w e e K im ²⁾

A b s trac t

W e fir st deriv e g r ou p or derin g r efer en ce prior s in a t w o - w ay m ix ed - effect s an aly sis of v arian ce (A N OV A ) m odel. W e sh ow t h at post erior distribut ion s are proper an d pr ov ide m ar gin al p ost erior dist ribu t ion s un der r efer en ce prior s . W e also ex am in e w h eth er t h e r eferen ce prior s s at isfy t h e pr ob ability m at chin g crit erion . F in ally , t h e r eferen ce prior satisfy in g th e pr ob ability m at chin g crit erion is sh ow n t o b e g ood in t h e sen se of fr equ ent ist cov era g e prob abilit y of th e post erior qu an tile .

K e y w o rd s : Error V arian ce, F r equ ent ist cov era g e pr ob ab ilit y , J effr ey s ' prior , M at ch in g prior , R eferen ce prior , T w o- w ay m ix ed - effect s A N OVA m odel.

1 . In tro du c ti on

T h e problem of est im at in g v arian ce in an aly sis of v arian ce m odel h a s b een inv est ig at ed by m any st atist ician s . E specially in Bay esian poin t of v iew , th e infer en ce of v arian ce com pon en t s in r an dom effect s m odel h a s b een t reat ed b y m an y st at istician s for a lon g tim e. F ir st , con sider t h e b alan ced on e - w ay r an dom effect m odel : y

_ij

= +

_i

+

_ij

, i = 1 , 2 , , I , j = 1, 2 , ,J , w h ere is an kn ow n con st ant s , an d t h e

i

an d

ij

ar e in depen dent n orm al v ariable s w ith 0 m ean s

1. Lect uring Pr ofes s or , Departm ent of Applied St atist ics , Konkuk Univ er sity , S eoul, 143- 701, Kor ea

E - m ail : ihchang @konkuk .ac.kr

2. Pr ofes s or , Departm ent of M athem at ics , Hany ang Univ er s ity , Seoul, 133- 791, Kor ea

an d v arian ce

²

an d

²

, r espectiv ely . B ox an d T iao (1973 ) ch ose a prior dist ribu tion ( ,

²

,

²

)

^{- 2}

(

²

+ J

²

)

^{- 1}

an d calcu lat ed t h e post erior dist ribu t ion s . T h ey ob s erv ed t h e r elat ion ship b et w een t h e post erior of = J

²

/

²

an d a fr equ ent ist r esu lt in a hy pot h esis t est in g problem . Y e (1994 ) dev elop ed r efer en ce prior s for , ex am in ed fr equ en tist cov er ag e pr ob abilities of post erior qu an tiles for v ariou s an d com p ar ed of th e Bay e s est im at or s for r efer en ce prior s . T h e or der ed g r ou p refer en ce prior alg orit hm of B er g er an d B ern ar d (1989 ) is applied t o t h e b alan ced on e - w ay r an dom effect m odel by Ber g er an d Bern ar do (1992a ). A lso, Ch un g an d Dey (1998 ) deriv ed r efer en ce prior s an d fir st order prob abilit y m at chin g prior s for int r acla s s corr elation =

²

/ (

²

+

²

) an d ex am in ed t h e fr equ ent ist cov er ag e pr ob ab ilit ies of post erior qu an tiles for v ariou s . Recent ly Kim , K an g , an d Lee (2001) deriv ed t h e secon d or der pr ob ability m at chin g crit erion for t h e r at io of th e v arian ce com pon en t s . T h ey sh ow s t h at am on g all of t h e r efer en ce prior s g iv en in Y e (1994 ), t h e on ly on e refer en ce prior s atisfie s t h e secon d or der m at ch in g crit erion .

N ow , con sider a t w o- w ay m ix ed - effect s an aly sis of v arian ce (A N OV A ) m odel:

y

_ij

=

_i

+

_j

+

_ij

, i = 1 , 2 , , p ( > 1) , j = 1 , 2 , , q( > 1) , ( 1 . 1) w h er e

_i

is t h e it h m ean effect of fix ed effect s , r an dom effect

_j

' s in depen dent an d iden tically distribut ed a s N ( 0 ,

²

) ,

_ij

' s ar e a s su m ed t o b e in depen den t an d ident ically dist ribu t ed a s N ( 0 ,

²

) . F urt h er , t h e

j

' s ar e also a s su m ed t o b e in dep en den t of

_ij

' s . V oun at su an d S m ith (1997) stu died Bay sian appr oach for v arian ce com pon ent m odel an d h ier ar ch ical m odel w it h 2- v arian ce com pon en t for b alan ced an d u nb alan ced ca se in t his m odel. Recent ly Ch an g an d Kim (2002) con sider t h e pr oblem of e st im at in g

²

in th is m odel (1.1) u sin g J effrey s ' prior , r efer en ce prior , an d m at chin g prior s . T h ey t h en com p ar e qu an tiles of m arg in al p ost erior den sit ies of

²

in t w o r eal dat a s et .

In t his paper , w e con sider a Bay e sian an aly sis of err or v arian ce in th e m odel

(1.1) u sin g r efer en ce prior s . S in ce ou r focu s is fu lly B ay esian , ch oice of prior s is

v ery im p ort ant . T h e det erm in at ion of r ea s on able n oninform at iv e prior s in

m u lt ip ar am et er pr oblem is n ot ea sy ; com m on n oninform at iv e prior s , su ch a s

J effery s ' prior can h av e featu r es th at h av e dr am at ic effect on t h e post erior . M or e

specifically , B ern ar do (1979 ) poin t ed ou t t h at if w e ar e int er est ed in a su b set of

p ar am et er s , t h e r est b ein g n uisan ce par am et er s , t h en J effrey s ' prior m ay b e

in appr opriat e for r epr es ent in g v a gu e of litt le prior inform at ion . In or der t o

ov er com e t his pr oblem , B ern ar do (1979 ) pr opos ed th e r efer en ce prior appr oach for

t h e dev elopm en t of t h e n oninform at iv e prior . Ber g er an d Bern ardo (1989, 1992b )

ex t en ded t h eir alg orit hm t o m ultipar am et er pr oblem .

T h e pu rpose in th is pap er is t o obt ain refer en ce prior s for (

²

,

²

, ) w h er e

2

is t h e p ar am et er of in t ere st . T h e paper is arr an g ed a s follow s . S ect ion 2 deriv es t h e r eferen ce prior s . In S ection 3, w e sh ow th at post erior dist ribu tion s ar e pr oper an d pr ov ide m ar g in al p ost erior dist ribu t ion s u n der r eferen ce prior s . In S ection 4, w e ex am in e w h et h er t h e r efer en ce prior s sat isfy th e pr ob ability m at ch in g crit erion . F in ally , S ection 5 pr ov ides fr equ en tist cov er ag e pr ob abilit ie s of t h e p ost erior cr edible set s u sin g t h e r efer en ce prior s .

2 . T h e Grou pe d Orde rin g Re f e ren c e P ri ors

In Ber g er an d Bern ardo ' s (1992a ) r efer en ce prior appr oach t o t h e on e - w ay r an dom - effect m odel, t h e or der ed g roup is v ery im port ant . T h at is , t h e form of r efer en ce prior s can b e ch an g ed b y t h e order ed gr ou pin g . T his div ides t h e p ar am et er s int o t w o sub g r ou p s , called par am et er of in t er est an d nu is an ce p ar am et er s . N ot ation su ch a s {

²

^{, ( ,}

²

⁾ } w ill b e u sed t o specify t h e gr ou p an d t h e im port an ce of p ar am et er s ; {

²

^{, ( ,}

²

⁾ } m ean s t h at th er e ar e t w o g r ou p s , w ith

2

b ein g m ore im port an t th an t h e g r ou p ( ,

²

) . A g r ou p su ch a s {

²

^{, ,}

²

}

m ean s th at

²

is t h e m ost im port an t par am et er an d

²

is t h e lea st im port ant . U sin g Ber g er an d Bern ar do ' s alg orit hm of com put in g t h e r eferen ce prior s , th e r efer en ce prior distribut ion s for differ en t g roup s of or derin g of (

²

, ,

²

) ar e obt ain ed a s follow s :

F or m odel (1.1), th e lik elih ood fun ction of par am et er s (

²

, ,

²

) is giv en b y l(

²

,

²

, | y ) (

²

)

^-

q( p - 1)

2

(

²

+ p

²

)

^-

q 2

ex p [ - 1 2 (

p

q

j = 1

( y

.j

- y

. .

)

²

2

+ p

²

+

p i = 1

q

j = 1

( y

ij

- y

i .

- y

.j

+ y

. .

)

²

2

+ q

p

i = 1

( y

i .

- y

. .

- (

i

-

.

) )

²

2

+ pq( y

. .

-

.

)

²

2

+ p

²

) ] , ( 2 . 1)

w h er e y

_{i .}

= 1 q

q

j = 1

y

_ij

, y

_.j

= 1 p

p

i = 1

y

_ij

, y

_{. .}

= 1 pq

p i = 1

q

j = 1

y

_ij

, an d

_.

= 1

p

p i = 1 i

.

N ow u sin g n ot at ion s y = ( y

11

, , y

pq

)

^T

an d = (

1

, ,

p

)

^T

, (2.1) b ecom es

l(

²

,

²

, | y ) (

²

)

- q( p - 1)

2

(

²

+ p

²

)

- q 2

ex p [ - 1 2 (

p

q

j = 1

( y

_.j

- y

_{. .}

)

²

2

+ p

²

+

p i = 1

q

j = 1

( y

_ij

- y

_{i .}

- y

_.j

+ y

_{. .}

)

²

2

) ]

ex p [ - q

2

²

( ( - y )

^'

( I

p

-

2

2

+ p

²

J

p

) ( - y ) ) ] . ( 2 .2)

F r om (2.2), F ish er in form ation m atrix is

I

₁

(

²

,

²

, ) =

q ( p - 1)

2

⁴

+ q

2 (

²

+ p

²

)

²

pq

2 (

²

+ p

²

)

²

0

^'

p q 2 (

²

+ p

²

)

²

p

²

q

2 (

²

+ p

²

)

²

0

^'

0 0 q

2

I

_p

- q

²

2

( p

²

+

²

) J

_p

, ( 2 . 3)

w h er e I

_p

is t h e p p iden tity m at rix an d J

_p

is th e p p m at rix w it h 1 a s all elem en t s .

F r om (2.3 ), w e h av e th e follow in g t h eor em of th e r efer en ce prior of th e or derin g g r ou p (

²

, ,

²

) .

T h e o re m 2 .1 . F or t h e b alan ced t w o - w ay m ix ed - effect s A N OV A m odel if

²

is t h e p ar am et er of int er est , t h en t h e r efer en ce prior dist ribu t ion s for differ ent gr oup s of or derin g for (

²

, ,

²

) are ;

Gr oup ord e ring R ef er en ce p r ior

{

²

^{, ( ,}

²

⁾ }

1

(

²

)

^-

1

2

(

²

+ p

²

)

^-

3 2

{ ⁽

²

^,

²

^{) ,} }{

²

^,

²

^, }{

²

^{, ,}

²

}

2

(

²

)

^{- 1}

(

²

+ p

²

)

^{- 1}

{ ⁽

²

^{, ) ,}

²

}

3

(

²

)

^-

p 2 - 3

4

(

²

+ p

²

)

^{- 1}

{ ⁽

²

^{, ,}

²

⁾ }

4

(

²

)

^-

p + 1

2

(

²

+ p

²

)

^-

3 2

P ro o f . W e apply an alg orith m by Ber g er an d Bern ar do (1989 ) t o (2.3 ). S in ce t h e deriv at ion s of ot h er refer en ce prior s ar e sim ilar , w e con sider only th e r efer en ce prior for t h e g roup { ⁽

²

^{, ) ,}

²

} ^.

S t ep 1 : T h e u su al r efer en ce prior for

²

w it h (

²

, ) g iv en b ecom es ,

(

²

| (

²

, ) ) = ( ^{2 (}

²

^{p + p}

²

^q

²

⁾

²

)

1 2

S t ep 2 : Ch oose an in cr ea sin g s equ en ce K

₁

K

₂

of com p act sub set s of t h e p ar am et er space for ( (

²

, ) ,

²

) su ch t h at

i

K

_i

= . T ak e K

_i

= [ 1

i , i] [ - i , i]

^p

[ 1

i , i] . T h en n orm alize (

²

| (

²

, ) ) on

_{i , (} ²_{, )}

= {

²

^{: ( (}

²

^{, ) ,}

²

^{) K}

i

} obt ain in g p

_i

(

²

|(

²

, ) ) = C

_i

( (

²

, ) ) (

²

| (

²

, ) ) w h er e

C

i^{- 1}

( (

²

, ) ) =

i 1 i

( p

²

q 2 )

1

2

(

²

+ p

²

)

^{- 1}

d

²

= ( p

²

q 2 )

1

2

1

p [ log (

²

+ p

_i

) - log (

²

+ p i ) ]

S t ep 3 : T h e m ar gin al r eferen ce prior for (

²

, ) w it h r esp ect t o p

_i

(

²

| (

²

, ) ) is

i

( (

²

, ) )

= ex p

{ ^{2 1}

¹ⁱⁱ

p ¹ [ log ( ⁽

²

+ p

²

^{+ p}

_i

) - log (

²

⁾

^{- 1 2}

+ p i ) ]

log q

^{p + 1}

( p - 1)

2 (

²

)

^{- p - 1}

(

²

+ p

²

)

^{- 1}

d

²

}

= [ q

^{( p + 1)}

( p - 1)

2 ]

1 2

(

²

)

^-

p + 1

2

e

^-

1

4[ log ( ²+ pi) + log ( ²+ p i) ]

S t ep 4 : T h e r efer en ce prior for ( (

²

, ) ,

²

) is for fix ed ( (

²₀

,

₀

) ,

²

0

) , ( (

²

, ) ,

²

) = lim

i

C

_i

(

²

, )

_i

(

²

, )

C

i

(

²0

,

0

)

i

(

²0

,

0

) (

²

|(

²

, ) )

= (

²

)

- p + 1 2

(

²0

)

^-

p + 1 2

e

- 1

4 log

2 2

0

( p

²

q

2 (

²

+ p

²

)

²

)

1 2

(

²

)

^-

p + 1

2 - 1

4

(

²

+ p

²

)

^{- 1}

= (

²

)

- p 2 - 3

4

(

²

+ p

²

)

^{- 1}

.

N ot e th at t h e prior dist ribu tion s ar e all fr ee of th e locat ion p ar am et er . A ll t h e refer en ce prior distribut ion s ar e pr oport ion al t o a n eg at iv e p ow er s of

²

an d

(

²

+ p

²

) . T h er efor e, a g en er al form of t h e prior can b e w rit t en a s

( ,

²

,

²

) (

²

)

^{- a}

(

²

+ p

²

)

^{- b}

, ( 2 .4 ) w h er e a an d b ar e p osit iv e n um b er s .

A lso, Jeffr ey s ' prior w hich is t h e s qu ar e root of t h e det erm in ant of t h e ex p ect ed F ish er in form ation m atrix is giv en b y

J

( ,

²

,

²

) (

²

)

^-

p + 1

2

(

²

+ p

²

)

^-

3

2

. ( 2 . 5)

T h er efor e, t h e J effr ey s ' prior is s am e a s t h e r efer en ce prior

₄

for (

²

, ,

²

) .

It h a s b een arg u ed t h e r eferen ce prior dist ribu t ion depen d on th e s am ple size

an d design an d t h er efore v iolat e th e lik elih ood prin ciple. H ow ev er , t h er e is n o

s at isfact ory m et h od of obt ain in g a n on in form ativ e prior in th is s cen ario.

3 . M arg in al P o s t e rior D i s tribu tion s

A ccor din g t o Bay e s t h eor em , t h e p ost erior dist ribu tion of (

²

, ,

²

) w it h r espect t o t h e prior s in (2.4 ) is g iv en by

(

²

, ,

²

| y ) (

²

)

- a - q ( p - 1)

2

(

²

+ p

²

)

- b - q 2

ex p [ - 1 2 (

p

q

j = 1

( y

. j

- y

. .

)

²

2

+ p

²

+

p i = 1

q

j = 1

( y

ij

- y

i .

- y

. j

+ y

. .

)

²

2

+ q

p

i = 1

( y

i .

- y

. .

- (

i

-

.

) )

²

2

+ p q( y

_{. .}

-

_.

)

²

2

+ p

²

) ] . ( 3 . 1) In t egr at in g out an d

²

, th e r esu lt in g m ar gin al post erior dist ribu tion of

²

is

(

²

| y ) (

²

)

- p ( q - 1)

2 - a - b + 1

e

- S

2 ²

(

1

0

w

q - 1 2 + b - 2

e

- S S 2 ² w

d w ) , ( 3 . 2) w h er e w =

2

2

+ p

²

. W e can see post erior distribut ion is pr oper fr om t h e follow in g t h eor em .

T h e o re m 3 .1 . T h e post erior dist rib ut ion s (3.1) is pr oper if a > 1 - ( p - 1)( q - 1) 2 an d b > 3 - q

2 .

P ro o f . W e prov e th e r esu lt for r eferen ce prior (2.4). F ull post erior dist ribu t ion s is g iv en by

R

(

²

, ,

²

| y ) [ C( y ) ](

²

)

- a - q ( p - 1)

2

(

²

+ p

²

)

- b - q 2

ex p [ - 1 2 (

p

q

j = 1

( y

_.j

- y

_{. .}

)

²

2

+ p

²

+

p i = 1

q

j = 1

( y

_ij

- y

_{i .}

- y

_.j

+ y

_{. .}

)

²

2

+ q

p

i = 1

( y

_{i .}

- y

_{. .}

- (

_i

-

_.

) )

²

2

+ pq( y

. .

-

.

)

²

2

+ p

²

) ] ,

w h er e

[ C( y ) ]

^{- 1}

=

0 0 R^p

(

²

)

- a - q ( p - 1)

2

(

²

+ p

²

)

- b - q 2

ex p [ - 1 2 (

p

q

j = 1

( y

_.j

- y

_{. .}

)

²

2

+ p

²

+

p i = 1

q

j = 1

( y

_ij

- y

_{i .}

- y

_.j

+ y

_{. .}

)

²

2

+ q

p

i = 1

( y

_{i .}

- y

_{. .}

- (

_i

-

_.

) )

²

2

+ p q( y

_{. .}

-

_.

)

²

2

+ p

²

) ] d d

²

d

²

. ( 3 .3 ) In t egr at in g out in (3.3 ), w e obt ain follow in g r esult

R^p

ex p [ - q 2

²

(

p

i = 1

( y

i .

- y

. .

- (

i

-

.

) )

²

+ p

²

2

+ p

²

( y

. .

-

.

)

²

) ] d

=

R^p

ex p [ - q

2

²

( ( - y )

^'

( I

_p

-

2

2

+ p

²

J

_p

) ( - y ) ) ] d

= ( 2 )

p 2

q

- p 2

(

²

)

p - 1

2

(

²

+ p

²

)

1

2

. ( 3 .4 )

T h en , u sin g th e re sult of (3.4 ), (3.3) is giv en by ( 3 . 3) = ( 2 )

p 2

q

^-

p 2

0 0

(

²

)

^-

q ( p - 1) - ( p - 1)

2 - a

(

²

+ p

²

)

^-

q - 1

2 - b

ex p [ - 1

2

²

( S +

2

2

+ p

²

S S ) ] d

²

d

²

, ( 3 . 5)

w h er e S =

p i = 1

q

j = 1

( y

ij

- y

i .

- y

.j

+ y

. .

)

²

is err or su m of s qu ar e,

S S = p

q

j = 1

( y

.j

- y

. .

)

²

is sum of squ ar es du e t o r an dom effect s . Let r = 1

2

, w =

2

2

+ p

²

, t h en

²

= 1 - w

p r w an d | J | = 1

p r

³

w

²

. T h er efor e, ( 3 . 5)

0 1

0

r

p q - p

2 + a + b - 3

w

q - 1 2 + b - 2

ex p [ - r

2 ( S + w S S ) ] d w d r

<

0

r

p q - p

2 + a + b - 3

e

- r 2 s

0

w

q - 1 2 + b - 2

e

- rS S

2 w

d w

0

r

( p - 1) ( q - 1 )

2 + a - 2

e

^-

r 2 s

d r

< if ( p - 1) ( q - 1)

2 + a - 1>0 an d q - 3

2 + b >0 .

R e m ark 3 .1 . T h e prior distribut ion w it h a = 1/ 2 in (2.4 ) pr odu ces an im pr oper p ost erior dist rib ut ion w h en p = 2 an d q = 2 . T his m ean s th at t h e prior

i

for t h e g r ou ped or derin g {

²

^{, ( ,}

²

⁾ } pr odu ces an im pr oper post erior dist ribu t ion w h en

p = 2 an d q = 2 .

4 . P ro b ability M at c hin g P rior

N ow w e ob t ain t h e pr ob abilit y m at ch in g prior s an d s ee w h et h er t h ey in clu de s am e of t h e r efer en ce prior s dev elop ed in S ect ion 2. F ir st , w e briefly rev iew t h e pr ob ab ilit y m at ch in g prior a s follow s .

F or a prior , let

1^{1 -}

( ; y ) den ot e th e ( 1 - ) th per cen tile of th e p ost erior dist ribu tion of

₁

, t h at is

P [

_{1 1}^{1 -}

( ; y ) | y ] = 1 - , (4 . 1)

w h er e = (

_{1 , 2}

,

₃

)

^T

an d

₁

is t h e param et er of in t ere st an d

₂

an d

₃

ar e n uisan ce param et er s . W e w ant t o fin d prior s for w hich

P [

_{1 1}^{1 -}

( ; y ) | ] = 1 - + o( n

^{- u}

) ( 4 .2 ) for som e u >0 , a s n g oes t o infinity . P rior s s atisfy in g (4.2) ar e called pr ob ab ilit y m at chin g prior s . If u = 1

2 , th en is r eferr ed t o a s a fir st or der m at ch in g prior , w hile if u = 1 , is r eferr ed t o a s a secon d or der m at chin g prior . In or der t o fin d su ch m at chin g prior s , it is con v en ien t t o in tr odu ce orth og on al p ar am etrization . T o th is en d , let

2

= ,

²

= 1

p ( - ) , = . ( 4 . 3)

W it h th is par am et rizat ion , b a s ed on t h e lik elih ood fun ction (2.2) an d F ish er inform at ion m at rix (2.3 ) of par am et er s (

²

,

²

, ) , t h e F ish er in form ation m atrix of p ar am et er s ( , , ) is g iv en by

I

₂

( , , ) =

q( p - 1)

2

²

0 0

^'

0 q

2

²

0

^'

0 0 q

I

p

- q( - )

p J

p

. (4 .4 )

T hu s is orth og on al t o an d in th e sen se of Cox an d Reid (1987). By T ib shirani (1989), t h e cla s s of fir st or der pr ob abilit y m at ch in g prior is ch ara ct erized b y

( 1)

M

( , , )

^{- 1}

g ( , )

an d th e fir st or der pr ob ability m at chin g prior of origin al param et er s (

²

,

²

, ) is g iv en by

( 1)

M

(

²

,

²

, ) (

²

)

^{- 1}

g (

²

+ p

²

, ) , (4 . 5)

w h er e g (

²

+ p

²

, ) is arbit r ary differ en t iable fun ction .

Clearly th e cla s s of prior g iv en in (4.5 ) is qu it e lar g e, an d it is im port ant t o n arr ow dow n t his cla s s of prior s . T h er efor e w e deriv e th e cla s s of s econ d or der pr ob ab ilit y m at ch in g prior s by M u k erj ee an d Gh osh (1997 ).

T h e o re m 4 .1 . T h e secon d order pr ob ability m at chin g prior s ar e giv en b y

( 2)

M

(

²

,

²

, ) (

²

)

^{- 1}

g (

²

+ p

²

, ) , (4 . 6)

w h er e g (

²

+ p

²

, ) is arbit r ary differ en t iable fun ction .

P ro o f . Let =

₁

, =

₂

,

_{i - 2}

=

_i

, i = 3 , , p + 2 , t h en secon d or der pr ob ability m at ch in g prior is of t h e form (4.5), an d also g m u st s atisfy an addition al differ en tial equ at ion a s follow s :

(

_M^{( 1)}

, ) =

2 2 1

( I

^-

1 2

11

g (

₂

, ,

_{p + 2}

) )

-

p + 2 r = 2

p + 2

s = 2 s

( L

_{11 r}

I

¹¹

I

^rs

I

1 2

11

g (

₂

, ,

_{p + 2}

) ) - 1

3

₁

( L

₁₁₁

( I

¹¹

)

²

I

1 2

11

g (

₂

, ,

_{p + 2}

) ) = 0 , (4 . 7)

w h er e

L

111

= E (

3

L

3 1

) = - q( p - 1)

3 1

+ 3 ( p - 1) ( q - 1)

3 1

- 3 ( p - 1)

3 1

,

L

₁₁₂

= L

₁₁₃

= = L

11( p + 2 )

= 0 ,

an d I

^ij

is ( i , j ) elem en t of

I

₂^{- 1}

=

2

²₁

q( p - 1) 0 0

^'

0 2

²2

q 0

^'

0 0

¹

q I

p

+

²

q J

p

.

T h en ev ery differ en t iable fun ction g s atisfie s (4.7 ). T h at is , all th e fir st or der pr ob ab ilit y m at chin g prior s ar e t h e secon d or der pr ob ab ilit y m at chin g prior s . T h u s t h e r esultin g secon d or der pr ob abilit y m at chin g prior s for ar e giv en by

( 2)

M

( , , )

^{- 1}

g ( , )

an d th e secon d or der pr ob abilit y m at chin g prior s for

²

ar e

( 2 )

M

(

²

,

²

, ) (

²

)

^{- 1}

g (

²

+ p

²

, )

T his com plet es t h e proof.

R e m ark 4 .1 . T h er e are in finit ely m an y m at chin g prior s for

²

up t o o( n

^{- 1}

) .

A m on g th e r eferen ce prior s dev eloped in S ection 2

2

is th e on ly m at chin g prior .

5 . S im u l ation an d D i s cu s s ion

W elch an d P eer s (1963) pr ov ed t h at , u sin g J effr ey s ' prior th e differ en ce b et w een t h e fr equ en t ist cov er ag e of t h e t h post erior qu ant ile an d is th e or der of

o( 1/ n ) w h en n g et s larg e. A popu lar pr opert y is t h at t h e frequ en tist cov era g e pr ob ab ilit y of ( 1 - ) th p ost erior qu an tile sh ow ed b e close t o 1 - .

T ab le 5.1 pr ov ides fr equ ent ist cov er ag e pr ob abilit ies of 0.05 (0.95 ) post erior qu ant ile for

²

u n der four r eferen ce prior s for differ en t v alu e of p an d q . T h e com pu t ation of th ose n um erical v alu es is b a sed on t h e follow in g alg orit hm for any fix ed tru e

²

an d pr esp ecified pr ob abilt y v alu e . H er e is 0.05 (0.95 ). Let (

²

) ( | y ) b e t h e p ost erior - qu an tile of

²

g iv en y . T h at is t o s ay , F ( (

²

) ( | y ) | y ) = , w h er e F ( | y ) is t h e m arg in al post erior distribut ion of

2

. T h en t h e fr equ en tist cov era g e pr ob ability of th is on e sided cr edib le in t erv al of

2

is

P

²

( :

²

) = P

²

( 0 <

²

(

²

) ( | y ) ) .

T a b l e 5 .1 . F re qu e n t i s t c o v e ra g e pro b ab ilit i e s o f 0 . 05 ( 0 . 95) p o s t e ri o r qu an t il e s f o r

²

( p , q)

1 2 3 4

(2, 8 ) 0.02 (0.92) 0.03 (0.93 ) 0.02 (0.92) 0.01(0.75 ) (3, 4 ) 0.03 (0.94 ) 0.05 (0.95 ) 0.05 (0.93 ) 0.03 (0.89 ) (4, 3 ) 0.05 (0.97 ) 0.04 (0.97 ) 0.01(0.80) 0.01(0.76 ) (5, 5 ) 0.05 (0.96 ) 0.05 (0.95 ) 0.02 (0.85 ) 0.01(0.73 ) (5, 6 ) 0.03 (0.93 ) 0.04 (0.94 ) 0.03 (0.90) 0.01(0.74 ) (6, 5 ) 0.04 (0.95 ) 0.05 (0.95 ) 0.03 (0.87 ) 0.01(0.77 ) (5, 10) 0.03 (0.90) 0.05 (0.95 ) 0.02 (0.85 ) 0.01(0.78 ) (8, 2) 0.03 (0.93 ) 0.06 (0.98 ) 0.02 (0.80) 0.01(0.70) (10,10) 0.03 (0.95 ) 0.04 (0.95 ) 0.01(0.85 ) 0.02 (0.85 )

T h e est im at ed P

²

( :

²

) w h en = 0 . 05 ( 0 . 95 ) is sh ow n in T able 5.1. A ctu ally

T able 5.1 w a s com pu t ed in t h e follow in g w ay . F or fix ed

²

, w e t ak e 10.000

in dep en den t r an dom sam ples of y fr om t h e m odel (1.1). N ot e th at u n der t h e prior

, for fix ed y ,

²

(

²

) ( | y ) if an d on ly if F ( (

²

) ( | y ) | y ) . U n der t h e prior , P

²

( :

²

) can b e est im at ed by th e relat iv e fr equ en cy of F ( (

²

) | y ) . F r om t h e T able 5.1,

₂

is th e m ost appealin g r eferen ce prior dist ribu tion in th e s en s e of th e a sy m pt otic fr equ ent ist cov er ag e pr ob ability .

R ef e re n c e s

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25, 318 - 329.

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[ 2002년 9월 접수, 2002년 10월 채택 ]