Supplement to the papers on “the polyserial correlation …emt-hogasa/epser_edsc_OUC.pdf · 2014....

115

Economic Review (Otaru University of Commerce),Vol.62, No.4, 115-164, March, 2012.

Supplement to the papers on “the polyserial correlation coefficients”and “discrepancy functions for general covariance structures”

Haruhiko Ogasawara

Parts A and B of this note are to supplement Ogasawara (2011, 2010),respectively.

Part A0. Derivation of the inverse expansion

Let2

0 0/ 'l L θ θ . Then, from (3.1) the inverse expansion of θ̂ is

3 41 1 2 1 3

0 0 02 30 0 0 0 0

2

1 1ˆ ˆ ˆ( ) ( )2 ( ') 6 ( ')

( )p

l l l

O N

θ θ L L θ θ L θ θθ θ θ θ θ

23

1 1 1

20 0 0 0

23 3

1 1 1 1

2 20 0 0 0 0 0

34

1 2

30 0 0

1

2 ( ')

1

2 ( ') ( ')

1( ).

6 ( ')p

l l l

l l l l

l lO N

L L Lθ θ θ θ

L L L Lθ θ θ θ θ θ

Lθ θ θ

Let E( ) L Λ I , where I is the information matrix per observation, and let

L Λ M , where1/2( )pO N M . Then, we obtain

1 1 1 1 1 1 1 3/2( )pO N L Λ Λ MΛ Λ MΛ MΛ .

The above results give

116

1/2

1

10

0 ( )

23

1 1 1 1

20 0 0 0

( )

23

1 1 1 1 1 1

20 0 0 0

ˆ

1E

2 ( ')

1E

2 ( ')

p

p

O N

O N

l

l l l

l l l

θ θ Λθ

Λ MΛ Λ Λθ θ θ θ

Λ MΛ MΛ Λ MΛ Λθ θ θ θ

31 1 1 1

20 0 0 0

23 3

1 1

2 20 0 0 0 0

E( ')

1E

2 ( ') ( ')

l l l

l l l

Λ Λ MΛ Λθ θ θ θ

Λ Λθ θ θ θ θ

3/2

23 3

1 1 1 1

2 20 0 0 0 0 0

34

1 1 2

30 0 0

( )

3( ) ( ) 2

01

1E E

2 ( ') ( ')

1E ( )

6 ( ')

( ),

p

p

O N

i ip

i

l l l l

l lO N

O N

Λ Λ Λ Λθ θ θ θ θ θ

Λ Λθ θ θ

Λ l

where ( )[ ]pO indicates that the sum of the terms in brackets is of order ( )pO

for clarity.

1. Expectations of the log likelihood derivatives

Let Bdiag( , )x zΛ Λ Λ and Bdiag( , )x zI I I , where Λ I ,

Bdiag( ) denotes the block diagonal matrix with the diagonal blocks being the

matrices in parentheses, and I is the information matrix per observation.

1.1 Information matrix

(1) μ and Σ

117

*1

1

*1 1

1

2 *1

2 *1 1 1 1

1

2 *

( ),

2[ { ( )} { ( )} ],

2

,'

2[( ) { ( )} ( ) { ( )} ],

2

1(2 )(2 )

4

Nx

ii

Nabx ab

i a i biab

x

Nx ab

a i b b i aiab

ac db adxab cd

ab cd

l

l

lN

l

l

Σ x μμ

Σ x μ Σ x μ

Σμ μ

Σ Σ x μ Σ Σ x μμ

1

41 1{ ( )} { ( )} ,

Ncb

i

aci d i b

Σ x μ Σ x μ

wherek

Σ denotes the sum of k terms with similar patterns. Then, we obtain2 * 2 *

1' '

'

( ) ( ) E ,' '

( ) 0,ab

x xx x

x

l lN N N

N

μμ μμ

μ

I Λ Σμ μ μ μ

I

2 *

( ) ( ) E (2 )(2 )4

( ),

ab cd ab cd

xx x ab cd

ab cd

ac db ad cb

l NN N

I Λ

that is,1

'( ) , ( ) 0,

1( ) (2 )(2 )( ),

4

( 1; 1)

ab

ab cd

x x

ac db ad cbx ab cd

r a b r c d

μμ μΛ Σ Λ

Λ

where, e.g., '( ) uv denotes the submatrix of the matrix in parentheses

corresponding to the product 'u v , and the ranges of (a, b), (c, d), and similar

pairs will be used hereafter in similar situations.

118

(2) 1 1( ( ,..., ) ')K β and 1( ( ,..., ) ')r ξ*

( 1) 11

1( )

i i i i

i

Nz

k z z k z zik z

l

, where

1

2

0 0 0

( ) ( ), ( ) ( ) ,

1( ) exp ,

22

' , ( 0),

z i

i i i i

i

i i

i i

z z z z

z

z z

z z i K K K

d

ξ x

*

11

( ),i i

i

Niz

z zi z

l

x

ξ

2 *

( 1) 1 11

( 1) 1 ( 1) 12

1( )

1( )( ) ,

i i i i i i

i

i i i i i i i i

i

Nz

k l z z z k l z z zik l z

k z z k z z l z z l z z

z

l

where i ik l z k l l z ,

2 *

( 1) 1 11

( 1) 1 12

2 *2

1 1 121

1( )

1( )( ) ,

1 1' ( ) ( ) ,

'

i i i i i i

i

i i i i i i

i

i i i i i i

i i

Nz

i k z z z k z z zik z

k z z k z z z z

z

Nz

i i z z z z z zi z z

l

l

xξ

x xξ ξ

119

2 *

|

2( 1)

2( 1)

1

( ) ( ) E E

1 1E

1 1

k l k l

zz z X Z X

k l

X k l k k k l k k k l k k l k l

k l

l k k l k l k

k k

lN N

N

I Λ

( 1) ( 1)2

1

1 1E ,

k l l k

X k l k k l

k k l k

N

where

1

/2 1/2

E {} {} ( ) ,

1 1( ) exp ( ) ' ( )

(2 ) | | 2

X

r

p d

p

x x

x x μ Σ x μΣ

,

2 *

|

2 2

1 1

1 1

( ) ( ) E E

1 1E ,

k k

zz z X Z X

k

k kX k k k k

k k k k

lN N

N

ξ ξI Λξ

x

2 *2

' ' | 11

1( ) ( ) E E E ' ( ) ,

'

Kz

z z X Z X X a aa a

lN N N

ξξ ξξI Λ xx

ξ ξ

that is,

( 1) ( 1)2

1

1 1

1

2' 1

1

1 1( ) =E ,

1 1( ) E ( ) ( ,

1( ) =E ' ( ) .

k l

k

k l l k

z X k l k k l

k k l k

z X k k k k k k

k k

K

z X a aa a

ξ

ξξ

Λ

Λ x

Λ xx

120

1.2 Expectations of the products of the log likelihood derivatives

Recall L Λ M and1 *l N l . Let Bdiag( , )x zM M M .

(1) μ and Σ

(1-1)

2

(2)0

0 0

'

E( ) E v( ) ' ,' '

l lN N

l Mθ θ (for 1 )

The nonzero results of2 2 ln ( ) ln ( )

E =E( ) ( ) ( ) ( ) ( ) ( )

X

x A x B x C x A x B x C

l l p pN

x x

θ θ θ θ θ θ , where

( )A denotes the A-th element of the vector in parentheses, are

21 1 1 1

2 8

2ln ( ) ln ( )E {( ) ( ) ( ) ( ) },

' 2

ln ( ) ln ( ) 1E (2 )(2 )(2 ){ }.

8

abX a b b a

ab

ac de fbX ab cd ef

ab cd ef

p p

p p

x xΣ Σ Σ Σ

μ μ

x x

(1-2)2 (1) 3

0E( )N l (for 3 )

The nonzero results of

3

( )EX

x

p

x

θ are

1 1

1 1

2ln ( ) ln ( ) ln ( )E {( ) ( )

' 2

( ) ( ) },

abX a b

ab

b a

p p p

x x xΣ Σ

μ μ

Σ Σ

121

3 3 3

15

8

ln ( ) ln ( ) ln ( ) 1E (2 )(2 )(2 )

8

(2

)

1(2 )(2 )(2 ) .

8

X ab cd ef

ab cd ef

ab cd ef ab cd ef cd ab ef ef ab cd

ab cd ef

ac de fbab cd ef

p p p

x x x

(1-3)2 (1) 2 (2)

0 0E( )N l l (for 3 )

2

31

E( ) ( ) ( ) ( )

ln ( ) ln ( ) ln ( ) ln ( )= E E ( ),

( ) ( ) ( ) ( )

x A x B x C x D

X X

x A x B x C x D

l l l lN

p p p pO N

θ θ θ θ

x x x x

θ θ θ θ

2 22

231

E E( ) ( ) ( ) ( ) ( ) ( ) ( )

ln ( ) ln ( ) ln ( ) ln ( )= E E ( ).

( ) ( ) ( ) ( ) ( )

x A x B x A x B x C x D x E

X X

x A x B x C x D x E

l l l l lN

p p p pO N

θ θ θ θ θ θ θ

x x x x

θ θ θ θ θ

(1-4)2 (2) (2)

0 0E( ')N l l (for 2 )

Define E{ E( )}X as E{ E( )}X X . In

2 22E E( ) E( ) ,

( ) ( ) ( ) ( ) ( ) ( )x A x B x C x D x E x F

l l l lN

θ θ θ θ θ θ

we derive the nonzero results of

122

2 2ln ( ) ln ( )E E ( ) E ( )

( ) ( ) ( ) ( )

E [{( ) ( ) }{( ) ( ) }]

E [{( ) ( ) }( ) ],

X X X

x A x B x C x D

X x AB x AB x CD x CD

X x AB x AB x CD

p p

x x

θ θ θ θ

L Λ L Λ

L Λ L

where

2 ln ( )( Bdiag( , )),

'x x z

x x

p

xL L L L

θ θ

2 2

2 2 4

ln ( ) ln ( )E E ( ) E ( )

ln ( ) ln ( ) 1=E (2 )(2 ) ,

4

X X X

ab c de f

ac bd efX ab de

ab c de f

p p

p p

x x

x x

2 2ln ( ) ln ( )E E ( ) E ( )X X X

ab cd ef gh

p p

x x

41 1

41 1

1(2 )(2 )(2 )(2 )

16

E 2 2 { ( )} { ( )}

2 2 { ( )} { ( )}

ab cd ef gh

ac db ad cb acX d b

eg hf eh gf egh f

Σ x μ Σ x μ

Σ x μ Σ x μ

161(2 )(2 )(2 )(2 ) ( ).

16ac eg dh bf df bh

ab cd ef gh

(1-5)2 (1) (2)

0 0E( ')N l l (for 2 )

For2 (1) 3

0E( )N l , see (1-2). The nonzero results of

123

ln ( ) ln ( )E {( ) ( ) }

( ) ( )

ln ( ) ln ( )E ( ) ( ) ( )

( ) ( )

X x AB x AB

x C x D

X x AB x AB x CD

x C x D

p p

p p

x xL Λ

θ θ

x xL Λ Λ

θ θ

are2 4ln ( ) ln ( ) ln ( ) 1

E = (2 )(2 )4

ca bd efX ab de

ab c de f

p p p

x x x,

2

4

ln ( ) ln ( ) ln ( )E E ( )

1= (2 )(2 ) ( ),

4

X X

ab cd e f

ac de bf df beab cd

p p p

x x x

2

4 8

ln ( ) ln ( ) ln ( )E E ( )

1= (2 )(2 )(2 )(2 ) .

16

X X

ab cd ef gh

ac de fg hbab cd ef gh

p p p

x x x

(1-6)2 (1) (3)

0 0E( ')N l l (for 2 )

For2 (1) 4

0E( )N l ,

22E E( )

( ) ( ) ( ) ( ) ( )x A x B x C x D x E

l l l lN

θ θ θ θ θ , and

2 22E E( ) E( )

( ) ( ) ( ) ( ) ( ) ( )x A x B x C x D x E x F

l l l lN

θ θ θ θ θ θ ,

see (1-3) and (1-4). The remaining results are

124

3

3

ln ( ) ln ( )E E ( )

( ) ( ) ( ) ( )

ln ( ) ln ( )E

( ) ( ) ( ) ( )

X X

x A x B x C x D

X

x A x B x C x D

p p

p p

x x

θ θ θ θ

x x

θ θ θ θ

in3

2E E( )( ) ( ) ( ) ( ) ( ) ( )x A x B x C x D x E x F

l l l lN

θ θ θ θ θ θ

First, we obtain the nonzero

3 ln ( )

( ) ( ) ( )x A x B x C

p

x

θ θ θ as

31 1 1 12ln ( )

{( ) ( ) ( ) ( ) }' 2

aba b b a

ab

p

xΣ Σ Σ Σ

μ μ ,

3 41 1

1 1

ln ( ) 1(2 )(2 ) [{ ( )} ( )

4

{ ( )} ( ) ],

acab cd b d

ab cd

d b

p

x

Σ x μ Σμ

Σ x μ Σ

and3

8 241 1

ln ( ) 1(2 )(2 )(2 )

8

{ ( )} { ( )} ,

ab cd ef

ab cd ef

ac de fb ac def b

p

x

Σ x μ Σ x μ

which yield the nonzero

3 ln ( ) ln ( )E

( ) ( ) ( ) ( )X

x A x B x C x D

p p

x x

θ θ θ θ as

3

41 1 1 1

ln ( ) ln ( )E

'

1(2 )(2 ) {( ) ( ) ( ) ( ) },

4

X

ab cd

acab cd d b b d

p p

x x

μ μ

Σ Σ Σ Σ

125

3

24

ln ( ) ln ( ) 1E (2 )(2 )(2 )(2 )

16

( ).

X ab cd ef gh

ab cd ef gh

ac de fg hb fh gb

p p

x x

(1-7)

33 (1) 4 (1) 2 2

0 0E( ) {E( )}N N N l l (for 4 )

2 3

3

ln ( ) ln ( ) ln ( ) ln ( )E E

( ) ( ) ( ) ( )

ln ( ) ln ( ) ln ( ) ln ( )E ( ) ( )

( ) ( ) ( ) ( )

X X

x A x B x C x D

X x AB x CD

x A x B x C x D

N N p p p p

N

p p p pN

x x x x

θ θ θ θ

x x x xΛ Λ

θ θ θ θ

3ln ( ) ln ( ) ln ( ) ln ( )=E ( ) ( )

( ) ( ) ( ) ( )X x AB x CD

x A x B x C x D

p p p p

x x x xΛ Λ

θ θ θ θ

(the fourth multivariate cumulant of ln ( ) / ( )x Ap x θ ’s).

The nonzero expectations required for the fourth cumulant are

8

ln ( ) ln ( ) ln ( ) ln ( )E

1(2 )(2 ) ( ) ,

4

X

ab cd e f

ef ac bd ad bc ae cf bdab cd

p p p p

x x x x

ln ( ) ln ( ) ln ( ) ln ( )E

1(2 )(2 )(2 )(2 )

16

X

ab cd ef gh

ab cd ef gh

p p p p

x x x x

126

6

4 15 105

3 ( )ab cd ef gh ab cd ef gh eg fh eh fg

ab cd ef gh ab cd ef gh

6 4 2

3 2 2

( , , , ) ( , ) ( , )

1(2 )(2 )(2 )(2 )

16ac de fg hb

ab cd ef gh

ac bd eg fh

ab cd ef gh c d g h

(note that

3ln ( ) ln ( ) ln ( ) ln ( )E ab cd

X

a b c d

p p p p

x x x xis

nonzero, but the corresponding cumulant is zero).

(1-8)3 (1) 3 (2)

0 0E( )N l l (for 4 )

The expectations are expressed as

10

( ) , where each ( ) is the product

of two expectations, which was shown in (1-2), (1-3), and (1-5).

(1-9)3 (1) 2 (2) 2

0 0E( )N l l (for 4 )

The results are expressed as

15

( ) , where each ( ) is the product of three

expectations, which was shown in (1-1) and (1-4).

(1-10)3 (1) 3 (3)

0 0E( )N l l (for 4 )


15


expectations, which was shown in (1-6).

(1-11)

3

E( ) ( ) ( )x A x B x C

l θ θ θ (for 1 and 3 )

127

The results are given by the Bartlett identity as follows:3 23ln ( ) ln ( ) ln ( )

E E( ) ( ) ( ) ( ) ( ) ( )

ln ( ) ln ( ) ln ( )E .

( ) ( ) ( )

x A x B x C x A x B x C

x A x B x C

p p p

p p p

x x x

θ θ θ θ θ θ

x x x

θ θ θ

(1-12)

4

E( ) ( ) ( ) ( )x A x B x C x D

l θ θ θ θ (for 2 and 4 )

As in (1-11), we obtain4 34

2 2 23 6

ln ( ) ln ( ) ln ( )E E

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

ln ( ) ln ( ) ln ( )E E

( ) ( ) ( ) ( ) ( ) ( )

ln ( ) ln ( ) lnE

( ) ( )

x A x B x C x D x A x B x C x D

x A x B x C x D x A x B

x C x D

p p p

p p p

p p p

x x x

θ θ θ θ θ θ θ θ

x x x

θ θ θ θ θ θ

x x

θ θ

( ) ln ( ) ln ( ) ln ( ).

( ) ( ) ( ) ( )x A x B x C x D

p p p

x x x x

θ θ θ θ

(2) β and ξ

(2-1)

2

(2)0

0 0

'

E( ) E v( ) ' ,' '

l lN N

l Mθ θ (for 1 )

The results are

128

2

( 1) ( 1) 2

2 21 1

( 1) ( 1) ( 1)

2 21 1

ln Pr( | ) ln Pr( | )E

E

k l m

zk z kz k z k

k l k k k l k

k k k k

z k l z mz k l z m

k l m

k k m m

Z Z

x x

2( 1)

1 ( 1) 12 2

2( 1)

1 ( 1) 12 21 1 1

E ( )

( ) ,

k lk k kX k l k k mk k m k k

k k k

k lk k kk l k k m k k mk k

k k k

2

( 1) ( 1) 2

2 21 1

( 1) ( 1)

12 211


E

( )

k l

zk z kz k z k

k l k k k l k

k k k k

Kz k l zaz k l

k l a aak k a

Z Z

x x

ξ

x

2( 1)

1 12 2

2( 1)

1 12 21 1 1

E ( )

( ) ,

k lk k kX k l k k k k

k k k

k lk k kk l k k k k

k k k

x

129

2

( 1)( 1)

1 12 21 1

( 1)

1


E ( ) ( )

k l

zk zk z kz k

k k k k k k k k

k k k k

zl z l

l

l l

Z Z

x x

ξ

x

1 12 21 1

( 1) 1 12

( 1) 1 121 1

1 1E ( ) ( )

1( )

1( ) ,

k k k kX k l k k k k k k k

k k k k

k kk l k k k k

k k

k kk l k k k k

k k

x

2

( 1)( 1)

1 12 21 1

11


'

E ' ( ) ( )

( )

k

zk z k z kz k

k k k k k k k k

k k k k

Kza

a aa a

Z Z

x x

ξ ξ

xx

1 12

1 121 1

1E ' ( ) ( )

1( ) ( ) ,

k kX k k k k k

k k

k kk k k k k

k k

xx

130

2

21 1 12

1

( 1)

1


'

1 1E ' ( ) ( )

k

K

m m m m m mm m m

z k z k

zm k

k k

Z Z

x x

ξ ξ

xx

21 1 12

21 1 12

1 1

1 1E ' ( ) ( )

1 1( ) ( ) ,

X k k k k k k

k k

k k k k k k k

k k

xx

2

21 1 12

1

11


1 1E ( ) ( )

( )

k l m

K

k l m a a a a a a zaa a a

Kzb

b bb b

Z Z

x x x

x x

21 1 1 12

1

1 1E ( ) ( ) ( )

K

X k l m a a a a a a a aa a a

x x x

.

(2-2)2 (1) 3

0E( )N l (for 3 )

( 1) ( 1) ( 1)

1 1 1

ln Pr( | ) ln Pr( | ) ln Pr( | )E

E

k l m

zk zl z mz k z l z m

k l m

k k l l m m

Z Z Z

x x x

131

33 2

( 1) 12 2 2( )1 1

32

( 1)( 1) 12( ) 1

1 1 1E

1,

X k l m k k l m k kk l mk k k

k l m k kk l m k

( 1) ( 1)

111 1

ln Pr( | ) ln Pr( | ) ln Pr( | )E

E ( )

k l

Kzk zl z az k z l

k l a aak k l l a

Z Z Z

x x x

ξ

x

2 21 12 2

1

2

( 1) 1 12( ) 1

1 1E ( ) ( )

1( ) ,

X k l k k k k k k

k k

k l k k k kk l k

x

2

( 1)

111

ln Pr( | ) ln Pr( | ) ln Pr( | )E

'

E ' ( )

k

Kzk zaz k

k a aak k a

Z Z Z

x x x

ξ ξ

xx

2 21 12 2

1

1 1E ' ( ) ( )X k k k k k k

k k

xx ,

3

31 12

1 1

ln Pr( | ) ln Pr( | ) ln Pr( | )E

1E ( ) E ( ) .

k l m

K Kza

k l m a a X k l m a aa aa a

Z Z Z

x x x x x x

x x x

(2-3)2 (1) 2 (2)

0 0E( )N l l (for 3 )

132

2

3

1

E( ) ( ) ( ) ( )

ln Pr( | ) ln Pr( | ) ln Pr( | ) ln Pr( | )= E E

( ) ( ) ( ) ( )

( )

z A z B z C z D

z A z B z C z D

l l l lN

Z Z Z Z

O N

θ θ θ θ

x x x x

θ θ θ θ

(see the expression of zΛ in 1.1 (2)),

22

23

1

E E( )( ) ( ) ( ) ( ) ( )

ln Pr( | ) ln Pr( | ) ln Pr( | ) ln Pr( | )= E E

( ) ( ) ( ) ( ) ( )

( )

z A z B z C z D z E

z A z B z C z D z E

l l l lN

Z Z Z Z

O N

θ θ θ θ θ

x x x x

θ θ θ θ θ

(see (2-1) and the expression of zΛ in 1.1 (2)).

(2-4)2 (2) (2)

0 0E( ')N l l (for 2 )

For2 (1) 2 (2)

0 0E( )N l l , see (2-3). Recall

2 ln Pr( | )

'z

z z

Z

xL

θ θ , then the

remaining results are E[{( ) ( ) }{( ) ( ) }]z AB z AB z CD z CD L Λ L Λ in

3 ln Pr( | ) ln Pr( | )E[{( ) ( ) }{( ) ( ) }]E .

( ) ( )z AB z AB z CD z CD

z E z F

Z Z

x xL Λ L Λ

θ θ

We obtain

2 2

( 1) ( 1) 2

2 21 1

ln Pr( | ) ln Pr( | )E ( ) ( )

E

k l m nz z

k l m n

zk zkz k z k

k l k k k l k

k k k k

Z Z

x xΛ Λ

133

( 1) ( 1)

2 21

z k l z k l

k l

k k

( 1) ( 1) 2

2 21 1

( 1) ( 1)

2 21

( ) ( )k l m n

zm z mz m z m

mn m m mn m

m m m m

zm n z m n

m n z z

m m

Λ Λ

2 22 2

1 1

2 221

( 1)( 1) 1 1( , )

1 1E

1

k kX k l mn k k k k

k k k k

k kk l m n k k k k

kl mn k k k

24

( 1) 12( )

2

( 1)( 1)( 1) 121 1

1

1

kk l m n k k k k

k l mn k k

kk l m n k k k k

k k

2 22 2 2 2

( 1) ( 1) 1 ( 1) ( 1) 13 3( ) ( ) 1

1 1( ) ( )

k l m nk l m n k k k lm n k k z zmn mnk k

Λ Λ ,

2 2

( 1) ( 1) 2

2 21 1

( 1) ( 1) ( 1)

2 21 1

ln Pr( | ) ln Pr( | )E ( ) ( )

E

k l mz z

k l m

z k zkz k z k

k l k k k l k

k k k k

zk l z mz k l z m

k l

k k m m

Z Z

ξ

x xΛ Λ

ξ

x

12( )

z m

m m m m m

m

134

( 1)

121

( ) ( ) ( )k l m

z m

m m m z z

m

ξΛ Λ

2

12

2

121 1 1

1E ( )

1( )

k m mX k l m k k m m m

k m m

k m mk k m m m

k m m

x

2

( 1) 121 1

2

( 1)( 1) 121

1( )

1( )

k m mkl m k k m m m

k m m

k m mk l m k k m m m

k m m

2

( 1) 1 12( )

2

( 1)( 1) 1 12( ) 1 1

1 1( )

1 1( ) ( ) ( ) ,

k l m

m mk l m k k m m m

k l k m m

m mk l m k k m m m z z

k l k m m

ξΛ Λ

2 2

'

( 1) ( 1) 2

2 21 1

( 1) ( 1)

2 21

ln Pr( | ) ln Pr( | )E ( ) ( )

'

E '

k lz z

k l

zk zkz k z k

k l k k k l k

k k k k

z k l z k l

k l

k k

Z Z

ξξ

x xΛ Λ

ξ ξ

xx

21 1 1 '2

1

1 1( ) ( ) ( ) ( )

k l

K

m m m m m m zm z zm m m

ξ ξΛ Λ

135

22

1 1 12

22

1 1 121 1 1

1 1E ' ( ) ( )

1 1( ) ( )

kX k l k k k k k k k k

k k k

kk k k k k k k k

k k k

xx

22

( 1) 1 1 1 12( )

'

1 1 1( ) ( )

( ) ( ) ,k l

k l k k k k k k k kk l k k k

z z

ξ ξΛ Λ

2 2

'

( 1) ( 1)

1 12 21 1

ln Pr( | ) ln Pr( | )E ( ) ( )

'

E ' ( ) ( )

k lz z

k l

z k z kz k z k

k k k k k k k k

k k k k

Z Z

ξ ξ

x xΛ Λ

ξ ξ

xx

( 1) ( 1)

1 12 21 1

'

( ) ( )

( ) ( )k l

zl zlz l z l

l l l l l l l l

l l l l

z z

ξ ξΛ Λ

2

1

2

1

1 1

1 1E ' ( )

1 1( )

X k l k k k k k

k k

k k k k k

k k

xx

2

( 1) 1 1 1 1 12( )

'

1 1 1( ) ( )

( ) ( ) ,k l

k l k k k k k k k k k kk l k k k

z z

ξ ξΛ Λ

136

2 2

'

( 1) ( 1)

1 12 21 1

ln Pr( | ) ln Pr( | )E ( ) ( )

'

E ' ( ) ( )

k lz z

k l

zl zlz l z l

k l l l l l l l l

l l l l

Z Z

x

ξ ξ

x xΛ Λ

ξ ξ

xx

21 1 1 '2

1

1 1( ) ( ) ( ) ( )

k l

K

m m m m m m z m z zm m m

ξ ξΛ Λ

1

21 1 1 12

1

21 1 1 '2

1 1

1E ' ( )

1 1 1( ) ( ) ( )

1 1( ) ( ) ( ) ( ) ,

k l

X k l l l l l

l

l l l l l l l l l l l

l l l

l l l l l l z z

l l

x

ξξ

xx

Λ Λ

2 2

2

21 1 12

1

ln Pr( | ) ln Pr( | )E ( ) ( )

1 1E ( ) ( )

( ) ( )

k l m n

k l m n

z z

k l m n

K

k l m n a a a a a a zaa a a

z z

Z Z

x x x x

x xΛ Λ

Λ Λ

2

21 1 1

1

1 1E ( )

( ) ( )k l m n

K

X k l m n a a a a a aa a a

z z

x x x x

Λ Λ

(2-5)2 (1) (2)

0 0E( ')N l l (for 2 )

For2 (1) 3

0E( )N l , see (2-2). The remaining results are

137

ln Pr( | ) ln Pr( | )E {( ) ( ) }

( ) ( )

ln Pr( | ) ln Pr( | )E ( ) ( ) ( ) .

( ) ( )

z AB z AB

z C z D

X z AB z AB z CD

z C z D

Z Z

Z Z

x xL Λ

θ θ

x xL Λ Λ

θ θ

We obtain the first term on the right-hand side of the above equation as follows:

2

( 1) ( 1) 2

2 21 1

( 1) ( 1) ( 1)

2 21 1

ln Pr( | ) ln Pr( | ) ln Pr( | )E

E

k l m n

zk z kz k z k

k l k k k

k k k k

z k l z m z nz k l z m

k l m

k k m m

Z Z Z

x x x

( 1)

1

z n

n

n n

2 2 2 2

2 21 1

22

( 1) 12( )

E

1

k k k kX k l mn k k k k

k k k k

kk l m n k k k k

mn k k

2

( 1)( 1) ( 1) 121 1

2 2 2 21 1

( 1)( 1) ( 1)( 1)2 21 1

1kk l m n k k k k

k k

k k k kk l m n k k k l mn k k

k k k k

2 2

3 2 2( 1) 1 ( 1) ( 1) 13 3

( ) ( )

3( 1)( 1)( 1) 13

1 1

1,

k l mn k k k l m n k kk l mnk k

k l m n k k

k

138

2

( 1) ( 1) 2

2 21 1

( 1) ( 1) ( 1)

2 21 1

ln Pr( | ) ln Pr( | ) ln Pr( | )E

E

k l m

z k z kz k z k

k l k k k

k k k k

z k l zm z az k l z m

k l m

k k m m

Z Z Z

x x x

ξ

x

11

( )K

a aa a

2

12

2 21

1 ( 1) 12 21 1

21

( 1)( 1) 121 1

E ( )

( ) ( )

( )

k kX k l m k k k k

k k

k k k kk k k k k l m k k k k

k k k k

k kk l m k k k k

k k

x

22 2

( 1) 1 1 ( 1)( 1) 1 13 3( )

1 1( ) ( )k l m k k k k k l m k k k k

k l k k

,

2

( 1) ( 1) 2

2 21 1

2

( 1) ( 1)

12 211

ln Pr( | ) ln Pr( | ) ln Pr( | )E

'

E '

( )

k l

zk zkz k z k

k l k k k

k k k k

Kzk l z az k l

k l a aak k a

Z Z Z

x x x

ξ ξ

xx

139

22

12

2 22 2

1 ( 1) 1 12 3( )1 1

1E ' ( )

1 1( ) ( ) ,

kX k l k k k k

k k

kk k k k k l k k k k

k lk k k

xx

2

( 1) ( 1)

1 12 21 1

( 1) ( 1)

1 1

ln Pr( | ) ln Pr( | ) ln Pr( | )E

E ( ) ( )

k l m

zk zkz k z k

k k k k k k k k

k k k k

zl z mz l z m

l m

l l m m

Z Z Z

x x x

ξ

x

2

1 2

2

1 21 1

1E ( )

1( )

kX k l m k k k k k

k k

kk k k k k

k k

x

2

( 1) 1 12( )

( 1) ( 1) 1 121 1

1 1( )

1 1( )

k l m k k k k k k kl m k k

k l m k k k k k k k

k k

21

( 1)( 1) 1 2

1( ) k

k l m k k k k k

k k

,

140

2

( 1) ( 1)

1 12 21 1

( 1)

111

ln Pr( | ) ln Pr( | ) ln Pr( | )E

'

E ' ( ) ( )

( )

k l

zk z kz k z k

k k k k k k k k

k k k k

Kzl z az l

l a aal l a

Z Z Z

x x x

ξ ξ

xx

1 12

1 121 1

1E ' ( ) ( )

1( ) ( )

kX k l k k k k k k k

k k

kk k k k k k k

k k

xx

1( 1) 1 12

1( 1) 1 12

1 1

1( ) ( )

1( ) ( ) ,

kk l k k k k k k k

k k

kk l k k k k k k k

k k

2

( 1) ( 1)

1 12 21 1

2

11

ln Pr( | ) ln Pr( | ) ln Pr( | )E

E ( ) ( )

( )

k l m

z k z kz k z k

l m k k k k k k k k

k k k k

Kz a

a aa a

Z Z Z

x x

x x x

ξ

x

141

21 12

21 12

1 1

1 1E ( ) ( )

1 1( ) ( ) ,

X l m k k k k k k k

k k

k k k k k k k

k k

x x

x

2

21 1 12

1

( 1) ( 1)

1 1

ln Pr( | ) ln Pr( | ) ln Pr( | )E

'

1 1E ' ( ) ( )

k l

K

a a a a a a zaa a a

zk zlz k z l

k l

k k l l

Z Z Z

x x x

ξ ξ

xx

22

1 1 12

22

1 1 121 1 1

22 1

( 1) 1 1 12( )

1 1E ' ( ) ( )

1 1( ) ( )

1 1( ) ( ) ,

kX k l k k k k k k

k k k

kk k k k k k

k k k

k kk l k k k k k k

k l k k k

xx

2

21 1 12

1

( 1)

111

ln Pr( | ) ln Pr( | ) ln Pr( | )E

'

1 1E ' ( ) ( )

( )

k l

K

l a a a a a a z aa a a

Kzk zbz k

k b bbk k b

Z Z Z

x

x x x

ξ ξ

xx

142

21 1 1 12

21 1 1 12

1 1 1

1 1E ' ( ) ( ) ( )

1 1( ) ( ) ( ) ,

kX l k k k k k k k k

k k k

kk k k k k k k k

k k k

x

xx

2

21 1 12

1

2

11

ln Pr( | ) ln Pr( | ) ln Pr( | )E

1 1E ( ) ( )

( )

k l m n

K

k l m n a a a a a a z aa a a

Kzb

b bb b

Z Z Z

x x x x

x x x

2 21 1 1 12

1

1 1E ( ) ( ) ( )

K

X k l m n a a a a a a a aa a a

x x x x

.

(2-6)2 (1) (3)

0 0E( ')N l l (for 2 )

For2 (1) 4

0E( )N l ,

22E E( )

( ) ( ) ( ) ( ) ( )z A z B z C z D z E

l l l lN

θ θ θ θ θ ,

and

2 22E E( ) E( )

( ) ( ) ( ) ( ) ( ) ( )z A z B z C z D z E z F

l l l lN

θ θ θ θ θ θ

see (2-3) an (2.4).The remaining results are

3

3

Pr( | ) Pr( | )E E( )

( ) ( ) ( ) ( )

Pr( | ) Pr( | )E

( ) ( ) ( ) ( )

z A z B z C z D

z A z B z C z D

Z ZN

Z Z

x x

θ θ θ θ

x x

θ θ θ θ

143

in

3

E E( )( ) ( ) ( ) ( ) ( ) ( )z A z B z C z D z E z F

l l l lN

θ θ θ θ θ θ .

First, we derive

3 Pr( | )

( ) ( ) ( )z A z B z C

Z

x

θ θ θ as follows:

3( 1) 2

1

( 1) ( 1)2 3

2 2 3 31 1

ln Pr( | )(1 )

3 2

z k z k

k l m k k

k l m k k

z k z kz k z k

k k k

k k k k

Z

x

32

( 1) 1 1 12 3( ) 1 1

2( 1)( 1) 1 12 3

1 1

1 2

1 2,

z k l m k k k k kk l m k k

z k l m k k k k k

k k

3( 1) 2

1

( 1)

1 12 21

ln Pr( | )(1 )

( ) ( )

z k z k

k l k k

k l k k

zk z k

k k k k k k

k k

Z

xx

ξ

( 1) ( 1)2 21 12 2 3 3

1 1

( 1) ( 1)

2 21

2 2 ( ) ( )

( )

z k z kz k z k

k k k k k k k

k k k k

zk l z k l

k l k l

k k

( 1) ( 1)

1 13 31

2 ( ) ( )z k l z k l

k k k k k l

k k

,

144

3( 1) 2

1

( 1)

1 12 21

ln Pr( | )' (1 )

'

( ) ( )

zk z k

k k

k k k

z k z k

k k k k k k

k k

Z

xxx

ξ ξ

2 21 1 12 3

( 1) ( 1)2 21 1 12 3

1 1

{2 ( ) } 2 ( )

{2 ( ) } 2 ( ) ,

zk z k

k k k k k k k k k

k k

z k z k

k k k k k k k k k

k k

3

2 21 1

1

31 1 1 12 3

ln Pr( | ) 1{( 1 ) (1 ) }

3 2( )( ) ( ) .

K

k l m a a a aak l m a

a a a a a a a a za

a a

Zx x x

x

Then, the expectations are

3( 1) 2

1

( 1) ( 1)2 3

2 2 3 31 1

ln Pr( | ) ln Pr( | )E E (1 )

3 2

zk z k

k l m k k

k l m n k k

zk zkz k z k

k k k

k k k k

Z Z

x x

32

( 1) 1 1 12 3( ) 1 1

( 1)2( 1)( 1) 1 12 3

1 1 1

1 2

1 2


zn z n

z k l m k k k k k n

k k n n

145

2 2 3

2 21 1

4

3 31

1 1 1 1=E (1 ) 3

1 12

X k l mn k k k k

k k k k

k

k k

3

2 3( 1) 1 1 12 3

( ) 1 1

2 2 2( 1) ( 1) 1 1 12 3

1 1

1 2

1 2

k l mn k k k k kk l m k k

k l m n k k k k k

k k

2 2 2( 1)( 1) 1 12 3

1 1

2 3( 1)( 1) ( 1) 1 12 3

1 1

1 2

1 2

k l mn k k k k k

k k

k l m n k k k k k

k k

2 2 3( 1) 12 3

2 2 3( 1)( 1)( 1) 12 3

1 1 1

1 3 2(1 )

1 3 2(1 ) ,

k l m n k k k k k k

k k k

k l m n k k k k k k

k k k

3( 1) 2

1

( 1) ( 1)2 3

2 2 3 31 1

ln Pr( | ) ln Pr( | )E E (1 )

3 2

z k z k

k l m k k

k l m k k

zk z kz k z k

k k k

k k k k

Z Z

x xx

ξ

146

32

( 1) 1 1 12 3( ) 1 1

2( 1)( 1) 1 1 12 3

11 1

1 2

1 2( )


Kza

z k l m k k k k k a aak k a

2 2 312 3

2 2 312 3

1 1 1

1 3 2=E (1 ) ( )

1 3 2(1 ) ( )

X k l m k k k k k k k

k k k

k k k k k k k

k k k

x

32

( 1) 1 1 12 3( ) 1 1

2( 1)( 1) 1 1 12 3

1 1

1 2

1 2( ) ,

k l m k k k k kk l m k k

k l m k k k k k k k

k k

3( 1) 2

1

( 1) ( 1) 21 12 2 2 2

1 1

ln Pr( | ) ln Pr( | )E E (1 )

( ) ( ) 2

z k z k

k l k k

k l m k k

zk zkz k z k

k k k k k k k k

k k k k

Z Z

x xx

ξ

( 1) 21 13 3

1

2 ( ) ( )z k z k

k k k k k

k k

2

( 1)

( 1) 1 1 12 3( ) 1

1 2( ) ( )

z m z m

zk l k k k k k k mk l k k m m

147

2 2

1

2 31 12 2 2 2

1 1

1 1=E (1 )

1 1 1 1( ) ( ) 2

X k l m k k

k k

k k k k k k k k

k k k k

x

31 13 3

1

1 12 ( ) ( )k k k k k

k k

2

2( 1) 1 1 12 3

( )

2( 1)( 1) 1 1 12 3

1 2( ) ( )

1 2( ) ( ) ,

k l m k k k k k kk l k k

k l m k k k k k k

k k

3( 1) 2

1

( 1) ( 1) 21 12 2 2 2

1 1

ln Pr( | ) ln Pr( | )E E ' (1 )

'

( ) ( ) 2

zk z k

k l k k

k l k k

zk z kz k z k

k k k k k k k k

k k k k

Z Z

x xxx

ξ ξ

( 1) 21 13 3

1

2 ( ) ( )z k z k

k k k k k

k k

2

( 1) 1 1 1 12 3( ) 1

1 2( ) ( ) ( )

Kz a

zk l k k k k k k a ak l ak k a

2 212 2

2 21 1 13 2

1 1

1 1 2=E ' (1 ) ( )

2 1 1( ) ( ) (1 ) ( )

X k l k k k k k k k k

k k k

k k k k k k k k k k k

k k k

xx

148

2 21 12 3

1 1

2 2( ) ( )k k k k k k k

k k

2

( 1) 1 1 1 12 3( )

1 2( ) ( ) ( )k l k k k k k k k k

k l k k

,

3( 1) 2

1

( 1)

1 12 21

ln Pr( | ) ln Pr( | )E E ' (1 )

'

( ) ( )

zk z k

k k

k l k k

z k z k

k k k k k k

k k

Z Z

x xxx

ξ ξ

2 21 1 12 3

( 1) ( 1)2 21 1 12 3

1 1

{2 ( ) } 2 ( )

{2 ( ) } 2 ( )

zk zk

k k k k k k k k k

k k

z k z k

k k k k k k k k k

k k

( 1)

1

zl z l

l

l l

2 2

1

21 12 2

1

1 1=E ' (1 )

1 1( ) ( )

X k l k k

k k

k k k k k k

k k

xx

3 2 2 21 1 12 3

3 2 2 21 1 12 3

1 1

1 1{2 ( ) } 2 ( )

1 1{2 ( ) } 2 ( )

k k k k k k k k k

k k

k k k k k k k k k

k k

149

2( 1) 1 1 12

2 21 1 1 1 12 3

1 1(1 ) ( )

1 2{2 ( ) } ( )

k l k k k k k k k k

k k


k k

2( 1) 1 1 12

1 1

2 21 1 1 1 12 3

1 1

1 1(1 ) ( )

1 2{2 ( ) } ( ) ,

k l k k k k k k k k

k k


k k

3( 1) 2

1

( 1)

1 12 21

ln Pr( | ) ln Pr( | )E E ' (1 )

'

( ) ( )

zk z k

l k k

k l k k

z k z k

k k k k k k

k k

Z Zx

x xxx

ξ ξ

2 21 1 12 3

( 1) ( 1)2 21 1 12 3

1 1

{2 ( ) } 2 ( )

{2 ( ) } 2 ( )

zk zk

k k k k k k k k k

k k

z k z k

k k k k k k k k k

k k

11

( )K

za

a aa a

212

2 21 1 1 12 3

1 1=E ' (1 ) ( )

1 2{2 ( ) } ( ) ( )

X l k k k k k k

k k


k k

x

xx

150

212

1 1

2 21 1 1 12 3

1 1

1 1(1 ) ( )

1 2{2 ( ) } ( ) ( ) ,

k k k k k k

k k


k k

3

2 21 1

1

( 1)31 1 1 12 3

1


1E {( 1 ) (1 ) }

3 2( )( ) ( )

k l m n

K

k l m a a a aa a

zn z n

a a a a a a a a z a n

a a n n

Z Z

x x x

x x

2 21 1

31 1 1 12 3

1E {( 1 ) (1 ) }

3 2( )( ) ( )

X k l m n n n n n

n

n n n n n n n n n n

n n

x x x

2 21 1

1

31 1 1 12 3

1 1

1{( 1 ) (1 ) }

3 2( )( ) ( ) ,

n n n n n

n

n n n n n n n n n n

n n

3 ln Pr( | ) ln Pr( | )E

k l m n

Z Z

x x

151

2 21 1

1

31 1 1 1 12 3

1

1E {( 1 ) (1 ) }

3 2( )( ) ( ) ( )

K

k l m n a a a aa a

Kzb

a a a a a a a a z a b bba a b

x x x x

2 21 1

1

31 1 1 1 12 3

1E {( 1 ) (1 ) }

3 2( )( ) ( ) ( ) .

K

X k l m n a a a aa a

a a a a a a a a a a

a a

x x x x

(2-7)

33 (1) 4 (1) 2 2

0 0E( ) {E( )}N N N l l (for 4 )

2 3

3


( ) ( )

ln Pr( | ) ln Pr( | ) ln Pr( | ) ln Pr( | )E E

( ) ( ) ( ) ( )

ln Pr( | ) ln Pr( | )( ) ( )

( ) ( )

z A z B

z C z D z A z B

z AB z CD

z C z D

N N Z Z

N

Z Z Z Z

Z ZN

x x

θ θ

x x x x

θ θ θ θ

x xΛ Λ

θ θ

3

ln Pr( | ) ln Pr( | ) ln Pr( | ) ln Pr( | )=E

( ) ( ) ( ) ( )

( ) ( )

z A z B z C z D

z AB z CD

Z Z Z Z

x x x x

θ θ θ θ

Λ Λ

(the fourth multivariate cumulant of ln Pr( | ) / ( )z AZ x θ ’s).

The expectations required in the above cumulants are

ln Pr( | ) ln Pr( | ) ln Pr( | ) ln Pr( | )E

k l m n

Z Z Z Z

x x x x

152

( 1) ( 1)

1 1

( 1) ( 1)

1 1

Ezk zlz k z l

k l

k k l l

zm z nz m z n

m n

m m n n

4

4 3( 1) 13 3 3

( )1

63 2 2

( 1)( 1)( 1) 1 ( 1)( 1) 13 3( )1

1 1 1E

1 1,

X k l mn k k l m n k kk l mnk k k

k l m n k k k l m n k kk l mnk k


k l m

Z Z Z Z

x x x x

ξ

( 1) ( 1)

1 1

( 1)

111

E

( )

zk zlz k z l

k l

k k l l

Kzm z az m

m a aam m a

x

3 31 13 3

1

32 2

( 1) 1 1 ( 1)( 1) 1 13 3( ) 1

1 1E ( ) ( )

1 1( ) ( ) ,

X k l m k k k k k k

k k

k l m k k k k k l m k k k kk l m k k

x


'k l

Z Z Z Z

x x x x

ξ ξ

2

( 1) ( 1)

111 1

E ' ( )K

zk zl zaz k z l

k l a aak k l l a

xx

153

2 2 21 13 3

1

22

( 1) 1 13( )

1 1E ' ( ) ( )

1( ) ,

X k l k k k k k

k k

k l k k k kk l k

xx


k l m n

Z Z Z Z

x x x x

3

( 1)

111

E ( )K

z k zaz k

l m n k a aak k a

x x x

3 31 13 3

1

1 1E ( ) ( ) ,X l m n k k k k k k

k k

x x x


k l m n

Z Z Z Z

x x x x

4

11

E ( )K

z a

k l m n a aa a

x x x x

413

1

1E ( ) ,

K

X k l m n a aa a

x x x x

(2-8)3 (1) 3 (2)

0 0E( )N l l (for 4 )

The expectations are expressed as

10

( ) , where each ( ) is the product

of two expectations (see (2-2), (2-3), and (2-5)).

(2-9)3 (1) 2 (2) 2

0 0E( )N l l (for 4 )

154


15


expectations (see (2-1) and (2-4) ).

(2-10)3 (1) 3 (3)

0 0E( )N l l (for 4 )

The results are similar to those in (2-9) (see (2-6)).

(2-11)

3

E( ) ( ) ( )z A z B z C

l θ θ θ (for 1 and 3 )

The results are given by the Bartlett identity (see (1-11)).

(2-12)

4

E( ) ( ) ( ) ( )z A z B z C z D

l θ θ θ θ (for 2 and 4 )

The results are given by the Bartlett identity (see (1-12)).

(3) , , ,μ Σ β and ξSome of the expectations of the log likelihood derivatives with respect to

xθ and zθ , e.g.,2E

( ) ( ) ( ) ( )x A x B z C z D

l l l lN

θ θ θ θ are nonzero

though the corresponding fourth cumulants are zero. The results are given as in(1-3), (1-4), (2-3), and (2-4).

2. The nonzero partial derivatives of η with respect to θ .

We define1/2 1( , ) (1 ' ) ( )S S R ξ Σ ξ Σξ , then

11( ' )K S τ β 1 ξ μ and

1/2 1(Diag ) S ρ Σ Σξ .

2.1 First derivatives

1 11

1 1 11

, ,

, ,

K k

k ab ab

k K k

k k k

SS S

SS S S

τ τ1 τ

τ τe 1 τ

155

where ke is the vector of an appropriate dimension whose k-th element is 1

and the remaining ones are 0.

3/2 1 1/2 1

1

2(Diag ) ( )

2 2

,

ab abaa aa ab ba

ab

ab

S S

SS

ρE Σξ Σ E E ξ

ρ

1/2 1 1(Diag ) ( ) ,k

k k

SS S

ρΣ Σ ρ

where12

2ab

a b

ab

SS

,

1( )k

k

SS

Σξ , and abE is the matrix of

an appropriate size whose (a, b)th element is 1 and the remaining ones are 0.

2.2 Second derivatives2

21

2 221 1

( , )

2 22 1 2

1

,

,

, ,

K k

ab k ab

ab cdab cd cd ab ab cd

k K k l l

k ab ab k l k

SS

S SS S

S SS S S

τ1

τ ττ

τ τe 1

2 21 1 1

2 2 221 1 1

( )

,

, ,

k ab k ab ab k k ab

k lk l l k k l l k k l

S S SS S S

S S SS S S

τ τ ττ

τ τ τ ττ

2 25/2 1 3/2

( , )

221 1 1

( , )

3(2 )

4 4

( ) ,

ababcd aa aa aa cd

ab cdab cd

ac ad ad acab cd cd ab ab cd

S

S SS S S

ρE Σξ

ρE E ξ ρ

156

23/2 1 1/2 1

21 1 1

2( ) (Diag ) ( )

2 2

,

ab abaa aa k a bk b ak

k ab

k ab ab k k ab

S S

S S SS S S

ρE Σ Σ e e

ρ ρρ

2 221 1

( )

,k lk l l k k l

S SS S

ρ ρ

ρ

where2

3

21 2

1(2 )(2 ) ,

4

2( ) ,

2

ab cd a b c d

ab cd

abk a b k b a a b

k ab k

SS

S SS S

2

1 1k l

k l k l

S S SS S

.

2.3 Third derivatives3 2 22

1 1

( , )

,ab cdab cd k cd k ab k ab cd

S SS S

τ τ τ

3 2 231 1

( , , )

31 ,

ab cd efab cd ef cd ef ab cd ab ef

ab cd ef

S SS S

SS

τ τ τ

τ

3 2 221 1

( , )

,ab cdk ab cd cd k ab k ab cd

S SS S

τ τ τ

3 2 2 21 1 1 ,

k ab l k l ab ab l k l k ab

S S SS S S

τ τ τ τ

157

3 221 2

( , )

21

ab cdk ab cd cd k ab cd k ab

cd ab k

S S SS S

SS

τ τ τ

τ

2 2 31 2 1 ,

k ab cd k ab cd k ab cd

S S S SS S S

ττ τ

3 2 2 21 1 1 ,

k l ab k l ab l ab k l k ab

S S SS S S

τ τ τ τ

3 2 221 1

( )

,k lk l m l m k m k l

S SS S

τ τ τ

3 221 2

( )

21

k lk l ab l ab k l ab k

l k ab

S S SS S

SS

τ τ τ

τ

2 2 31 2 1 ,

ab k l ab k l k l ab

S S S SS S S

ττ τ

3 2 221 1

( )

,k lk l m l m k m k l

S SS S

τ τ τ

3 2 231 1

( )

31 ,

k l mk l m l m k l k m

k l m

S SS S

SS

τ τ τ

τ

3 37/2 1 5/2

( , , )

2 21 1 1

15 3(2 )

8 8

( )

abcdef aa aa abcd aa efab cd efab cd ef

ae af af ae

cd ef ab ef ab cd

S

S SS S S

ρ

E Σξ

ρ ρE E ξ

158

31 ,

ab cd ef

SS

ρ

3 25/2 1 2 3/2

( , )

1 2

3( )

4 4

(2 ) ( ) ( )

ababcd aa aa k aa

ab cdk ab cd k

cd ac d k ad ck a ac ad ad ac

k

SS S

SS S

ρ

E Σ Σξ

e E E ξ

2 221 2 1

( , )

2 2 31 2 1 ,

ab cd k cd ab cd k ab cd k ab

k ab cd k ab cd k ab cd

S S S SS S S

S S S SS S S

ρ ρ ρ

ρρ ρ

3 221 2

( )

21

k lk l ab l ab k l ab k

l k ab

S S SS S

SS

ρ ρ ρ

ρ

2 2 31 2 1 ,

ab k l ab k l k l ab

S S S SS S S

ρρ ρ

3 2 2 331 1 1

( )k l mk l m l m k l k m k l m

S S SS S S

ρ ρ ρ

ρ ,

where3

53(2 )(2 )(2 ) ,

8ab cd ef a b c d e f

ab cd ef

SS

3 23

( , )

4

1(2 )(2 ) ( )

4

3,

4

ab cd k a b kb a c dab cdk ab cd

a b c d

k

SS

SS

159

31 2

221 2

( )

2( )

2

,

abk a lb kb l a k l

k l ab ab

k l k ab l k l ab

S SS S

S S S S SS S

3 231

( )k l mk l m k m l

S S SS

.

3. Computation by Gaussian quadrature3.1 Univariate case

Stroud and Sechrest (1966, Table Five, pp.217-252) gave the following

values of iA and ix :

2

1

exp( ) ( ) ( )n

i ii

x f x dx A f x

,

where n=2(1)64(4)96(8)136. Let 2 .y x Then,

2

2

1

1( ) ( ) exp( / 2) ( )

2

1 1exp( ) ( 2 ) ( 2 ).

n

i ii

y f y dy y f y dy

x f x dx A f x

The above result corresponds to Bock and Lieberman (1970, Equation (5)).

Bock and Lieberman (1970, p.183) used n=64, where only 40 values of ix

were employed since iA ’s for the remaining ix ’s are1410iA . Bock and

Aitkin (1981, Table 1) reported the results using n=10 and 2.

3.2 Bivariate case

Suppose that ( , ) ' ( , )X Y NU μ Σ . Let the density of U at ( , ) 'x yube

12 1/2

1 1( , ) exp ( ) ' ( )

2 | | 2x y

u μ Σ u μΣ ,

160

where

2

2, ( , ) '

x xy

x y

yx y

Σ μand 2 2( ) ( , )x y U u

2( , , , )x y μ Σ . Define

22

1 2

1 ( )( , , ) exp

22x

xx

xx

(note

1( ,0,1) ( )x x ). Then, using the transformations

xx

x

xz

and

2|

| 2 2 2 1/2|

{( / )}( )

[ {( ) / }]

y xy x x y x

y x

y xy x y x

y x yz

, it follows

that

2

21

2 2 2 21

| | | | |

( , ) ( , )

( , , )

[ , ( / )( ), {( ) / }] ( , )

( ) ( ) ( , )

x x

y xy x x y xy x

x y x x x x y x y x y x x y x

x y g x y dxdy

x

y x g x y dxdy

z z g z z dz dz

| |1

*| |

1 1

1( ) ( , 2 )

1( 2 , 2 ),

n

x j x x x y x y x j xj

n n

i j x x i y x y x ji j

z A g z x dz

A A g x x

where*| | 2

( / ) 2x x i

y x y xy x iy xx

.

161

Part B

1. The partial derivatives of ULSF with respect to θ̂

2 2ULS ULS

2ULS

3 3 23ULS

ˆ ˆ ˆ ˆˆ ˆtr ( ) , tr ( ) ,

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ(2 ) ,

ˆ ˆ

ˆ ˆ ˆˆtr ( )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ

i i i j i j i j

ab

i ab i ab

i j k i j k i

F F

F

s

F

Σ Σ Σ ΣΣ S Σ S

Σ

Σ Σ ΣΣ S

,

ˆ ˆj k

3 2ULS

ˆ(2 ) ,

ˆ ˆ ˆ ˆab

i j ab i j ab

F

s

Σ

4 4ULS

3 2 24 3

ˆˆtr ( )


ˆ ˆ ˆ ˆ,


i j k l i j k l

i j k l i j k l

F

ΣΣ S

Σ Σ Σ Σ

4 3ULS

ˆ(2 ) ,

ˆ ˆ ˆ ˆ ˆ ˆab

i j k ab i j k ab

F

s

Σ

5 5ULS

4 2 35 10

ˆˆtr ( )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ,

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

i j k l m i j k l m

i j k l m i j k l m

F

ΣΣ S

Σ Σ Σ Σ

162

5 4ULS

ˆ(2 )


( , , , , 1,..., ; 1).

ab

i j k l ab i j k l ab

F

s

i j k l m q p a b

Σ

Using the above results with Lemma 1, we have the partial derivatives of

ULSF̂ with respect to s as follows:

ULS

2 2ULS

2 2

ˆˆ ˆˆ ˆtr ( ) (2 )( ) ,

ˆ

ˆˆˆ ˆ ˆ ˆˆtr +( )

ˆ ˆ ˆ ˆ

ˆ ˆˆ ˆˆtr ( ) (2 )ˆ ˆ

iab ab

ab abi

ji

ab cd ab cdi j i j

i ab iab

ab cdi i

F

s s

F

s s s s

s s

ΣΣ S Σ S

Σ Σ ΣΣ S

ΣΣ S

(2 ) ,

cd

ab ac bd

s

3 2 33ULS

22 3

3

ˆˆ ˆˆ ˆ ˆ ˆˆtr +( )

ˆ ˆ ˆ ˆ ˆ ˆ

ˆˆˆ ˆ ˆˆtr +( )

ˆ ˆ ˆ ˆ

ˆˆˆtr ( )

ˆ

ji k

ab cd ef ab cd efi j k i j k

ji

ab cd efi j i j

i

i

F

s s s s s s

s s s

Σ Σ ΣΣ S

Σ Σ ΣΣ S

ΣΣ S

ab cd efs s s

2 23 ˆˆ ˆˆ ˆ(2 ) ,

ˆ ˆ ˆjab i ab i

ab

cd ef cd efi j is s s s

4 2 2 33 4ULSˆ ˆ ˆ ˆ ˆ

trˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ab cd ef gh i j k l i j k l

F

s s s s

Σ Σ Σ Σ

163

4

22 33 6

ˆˆ ˆ ˆˆˆ+( )

ˆ ˆ ˆ ˆ

ˆˆ ˆˆ ˆ ˆˆtr +( )

ˆ ˆ ˆ ˆ ˆ ˆ

ji k l

ab cd ef ghi j k l

ji k

ab cd ef ghi j k i j k

s s s s

s s s s

ΣΣ S

Σ Σ ΣΣ S

2223 3

3 44

ˆˆˆ ˆ ˆˆtr +( )

ˆ ˆ ˆ ˆ

ˆˆ ˆˆˆtr ( )

ˆ

ji

ab cd ef ghi j i j

ji i

ab cd ef gh ab cd ef ghi

s s s s

s s s s s s s s

Σ Σ ΣΣ S

ΣΣ S

23 24 3

3

ˆ ˆˆ ˆ ˆˆ ˆ(2 )

ˆ ˆ ˆ ˆ ˆ

ˆˆ

ˆ

( 1, 1, 1, 1),

j jab i k ab iab

cd ef gh cd ef ghi j k i j

ab i

cd ef ghi

s s s s s s

s s s

p a b p c d p e f p g h

where Einstein’s summation convention is to be used only for subscripts i, j, kand l.

2. The chain rules

2 2 2

2

ˆˆ ˆ,

ˆ

ˆ ˆ ˆˆ ˆ ˆ,

ˆ ˆ

ab ab

ab cd ab cd ab cd

g g F

s sF

g g F F g F

s s s s s sF F

164

3 3

3

2 2 33

2

ˆ ˆ ˆˆ ˆ

ˆ

ˆ ˆ ˆˆ ˆ,

ˆ ˆ

ab cd ef ab cd ef

ab cd ef ab cd ef

g g F F F

s s s s s sF

g F F g F

s s s s s sF F

4 4

4

3 2 2 2 26 3

3 2

2 3 44

2

ˆ ˆ ˆ ˆˆ ˆ

ˆ

ˆ ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ

ˆ ˆ ˆˆ ˆ

ˆ ˆ

( 1,




g g F F F F

s s s s s s s sF

g F F F g F F

s s s s s s s sF F

g F F g F

s s s s s s s sF F

p a b p

1, 1, 1).c d p e f p g h

ReferencesBock, R. D., & Lieberman, M. (1970). Fitting a response model for n

dichotomously scored items. Psychometrika, 35, 179-197.Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of

item parameters: Application of an EM algorithm. Psychometrika, 46,443-459.

Ogasawara, H. (2011). Asymptotic expansions of the distributions of thepolyserial correlation coefficients. Behaviormetrika, 38 (2), 153-168.

Ogasawara, H. (2010). Asymptotic expansions of the null distributions ofdiscrepancy functions for general covariance structures undernonnormality. American Journal of Mathematical and ManagementSciences, 30 (3 &4), 385-422.

Stroud, A. H., & Secrest, D. (1966). Gaussian quadrature formulas.Englewood Cliffs, NJ: Prentice-Hall.

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