conomic Review (Otaru University of Commerce Vo.6 No.4 5-64 March 0. Suppement to the papers on the poyseria correation coefficients and discrepancy functions for genera covariance structures Haruhio Ogasawara Parts A and B of this note are to suppement Ogasawara (0 00 respectivey. Part A 0. Derivation of the inverse expansion Let L / θ θ '. Then from (. the inverse expansion of ˆθ is 0 0 ˆ ˆ θθ L L θθ L θˆ θ 4 0 ( 0 ( 0 θ0 θ0( θ0 ' 6 θ0 ( θ0 ' O ( p N L L L θ0 θ0( θ0 ' θ0 L L L L θ0( θ0 ' θ0 θ0( θ0 ' θ 0 4 O p N L 0 0 0 6 θ ( θ ' θ (. Let ( L Λ I where I is the information matrix per observation and et L Λ M where M O N ( / p. Then we obtain The ove resuts give / L Λ Λ M Λ Λ M Λ M Λ O N. p ( 5
ˆ θ θ0 Λ θ 0 / Op ( N Λ MΛ Λ Λ θ0 θ0( θ0 ' θ0 Λ MΛ MΛ Λ MΛ Λ θ0 θ0( θ0 ' θ0 Λ Λ MΛ Λ θ0( θ0 ' θ0 θ0 Λ Λ θ0( θ0 ' θ0( θ0 ' θ0 i 4 O p N Λ θ0 θ0 θ0 / O ( N ( i ( i 0 p p O ( N Λ Λ Λ Λ θ0( θ0 ' θ0 θ0( θ0 ' θ 0 Λ ( 6 ( ' Λ O ( N p where [ ] O p ( indicates that the sum of the terms in bracets is of order Op( for carity.. xpectations of the og ieihood derivatives Let Λ Bdiag( Λx Λ and I Bdiag( Ix I where Λ I Bdiag( denotes the boc diagona matrix with the diagona bocs being the matrices in parentheses and I is the information matrix per observation.. Information matrix ( μ and Σ 6
* x μ * x N Σ xi μ i ( * x N Σ μμ' * x μ N [ { Σ ( xi μ} a{ Σ ( xi μ} b] i N [( Σ a{ Σ ( xi μ} b ( Σ b{ Σ ( xi μ} a] i * N x ( ( cd cd 4 i where 4 7 ac db ad Σ x μ Σ x μ ac { ( i } d{ ( i } b Σ denotes the sum of terms with simiar patterns. Then we obtain N N N μμ' μμ' N( I 0 * * x x ( Ix μμ' ( Λx μμ' Σ x μ' * x N N( Ix ( ( ( N Λ cd x cd cd cd 4 that is ( Λx μμ' Σ ( Λx μ 0 ac db ad cb ( Λx ( ( ( cd cd 4 ( r a b ; r c d where e.g. ( uv ' cb ac db ad cb ( denotes the submatrix of the matrix in parentheses corresponding to the product pairs wi be used hereafter in simiar situations. u v ' and the ranges of (a b (c d and simiar
( β( (... ' and ξ ( ( (... r ' * N i i ( i i i where i ( ( ( i ( d i i i i i ( exp i i ξ ' x ( 0 i 0 0 0 i ξ * i N i x i i ( i i * N ( i i i ( i i i i i where i i * ξ ( ( i i ( i i i i ( i i i N xi ( i i i ( i i i i i ( ( i i ( i i i i i * ξξ' i i N xixi ' ( ( i i i i i i i 8
* N( I ( N Λ X Z X N X ( ( N where ( ( X {} {} p( d p X x x ( Σ ( x exp ( ' ( r/ / x μ Σ x μ * N( I ξ N( Λ ξ X Z X ξ N X x * N( I ξξ' N( Λ ξξ' XZ X N X xx' ( a a ξξ' a a that is ( ( ( Λ = X ( Λ ξξ' = X xx' ( a a. a a 9 ( Λ ξ ( ( X x
. xpectations of the products of the og ieihood derivatives * Reca L Λ M and N. Let M Bdiag( Mx M. ( μ and Σ ' ( ( N (- 0 N v( M' θ0 ' θ 0 ' (for The nonero resuts of n p( n p( N = x x X ( θx A( θx B ( θx C ( θx A( θx B ( θ x where C ( A denotes the A-th eement of the vector in parentheses are n p( x n p( x X Σ a Σ b Σ b Σ a μ μ' {( ( ( ( } 8 n p( x n p( x ac de fb X ( ( cd ( ef { }. cd ef 8 (- ( N (for ( 0 The nonero resuts of X p( x θ x n p( x n p( x n p( x X a b {( Σ ( Σ μ μ' are ( Σ b( Σ a} 0
n p( x n p( x n p( x X ( ( cd ( ef cd ef 8 cd ef cd ef cd ef ef cd ( 5 cd ef 8 ( ( ( ac de fb cd ef. 8 (- N ( ( ( N 0 0 (for ( θ ( θ ( θ ( θ x A x B x C x D n p( x n p( x n p( x n p( x = X X O( N ( θx A ( θx B ( θx C ( θx D N ( θ ( θ ( θ ( θ ( θ ( θ ( θ n p( x n p( x n p( x n p( x = X X O( N. ( θx A( θx B ( θx C ( θx D ( θx (-4 x A x B x A x B x C x D x ( ' ( ( N 0 0 (for Define { X ( } as { X ( X } N ( ( ( θx A( θx B ( θx C ( θx D ( θx ( θx F we derive the nonero resuts of. In
n p( x n p( x X X ( X ( ( θx A( θx B ( θx C ( θx D [{( L ( Λ }{( L ( Λ }] X x AB x AB x CD x CD [{( L ( Λ }( L ] where X x AB x AB x CD n p( x Lx ( L Bdiag( Lx L θ θ ' x x n p( x n p( x X X ( X ( c de f 4 n p( x n p( x ac bd ef = X ( ( de c de f 4 n p( x n p( x X X ( X ( cd ef gh ( ( cd ( ef ( gh 6 4 ac db ad cb ac X { ( } d{ ( } b Σ x μ Σ x μ 4 eg hf eh gf eg { Σ ( x μ} h{ Σ ( x μ} f 6 ( ( ( ( ac eg ( dh bf df bh cd ef gh. 6 (-5 For ( ' ( ( N 0 0 (for ( N ( 0 see (-. The nonero resuts of
n p( x n p( x X {( Lx AB ( Λx AB} ( θx C ( θx D n p( x n p( x ( L ( Λ ( Λ ( θx C ( θx D are X x AB x AB x CD 4 n p( x n p( x n p( x X = ( ( de c de f 4 n p( x n p( x n p( x X X ( cd e f 4 ac de bf df be = ( ( cd ( 4 n p( x n p( x n p( x X X ( cd ef gh 4 8 ac de fg hb = ( ( cd ( ef ( gh. 6 ca bd ef (-6 For ( ' ( ( N 0 0 (for ( N ( 4 0 ( θ x A( θ x B ( θ x C ( θ x D ( θ x and N ( N ( ( ( θx A( θx B ( θx C ( θx D ( θx ( θ x F see (- and (-4. The remaining resuts are
n p( x n p( x X X ( ( θx A( θx B( θx C ( θx D n p( x n p( x X ( θx A( θx B( θx C ( θx D in ( θ ( θ ( θ ( θ ( θ ( θ N ( x A x B x C x D x x F First we obtain the nonero n p( x μμ' n p( x ( θ ( θ ( θ x A x B x C {( Σ a( Σ b ( Σ b( Σ a } 4 n p( x ( ( ac [{ ( } ( cd Σ x μ b Σ d cd μ 4 and n p( x ( ( cd ( ef 8 cd ef as { Σ ( x μ} d ( Σ b] n p( x n p( x X ( θx A( θx B( θx C ( θx as D n p( x n p( x X cdμ μ' 4 ( ( ac {( ( ( ( cd Σ d Σ b Σ b Σ d } 4 which yied the nonero 8 4 ac de fb ac de { Σ ( x μ} f { Σ ( x μ} b 4
n p( x n p( x X ( ( cd ( ef ( gh cd ef gh 6 4 ac de fg hb fh gb (. (-7 ( 4 ( 0 0 ( N N {( N } (for 4 n ( x n ( x n ( x n ( x N N p p p p X X N ( θx A ( θx B ( θx C ( θx D n p( x n p( x n p( x n p( x X x AB x CD N( Λ ( Λ ( θx A ( θx B ( θx C ( θx D n p( x n p( x n p( x n p( x = X x AB x CD ( Λ ( Λ ( θx A ( θx B ( θx C ( θx D (the fourth mutivariate cumuant of n p( x / ( θ s. X The nonero expectations required for the fourth cumuant are n p( x n p( x n p( x n p( x cd e f 8 ( ( ef ( ac bd ad bc ae cf bd cd 4 n p( x n p( x n p( x n p( x X cd ef gh ( ( cd ( ef ( gh 6 x A 5
6 cd ef gh cd ( ef gh eg fh eh fg 4 5 05 cd ef gh cd ef gh 64 ( ( cd ( ef ( gh 6 (note that X ac de fg hb ac bd eg fh ( cd ef gh ( c d ( g h n p( x n p( x n p( x n p( x a b c d nonero but the corresponding cumuant is ero. cd is (-8 ( ( ( N 0 0 (for 4 The expectations are expressed as 0 ( where each ( of two expectations which was shown in (- (- and (-5. is the product (-9 ( ( ( N 0 0 (for 4 The resuts are expressed as 5 ( where each ( expectations which was shown in (- and (-4. (-0 ( ( ( N 0 0 (for 4 The resuts are expressed as expectations which was shown in (-6. 5 ( where each ( is the product of three is the product of three (- ( θx A ( θx B ( θ x (for C and 6
The resuts are given by the Bartett identity as foows: n p( x n p( x n p( x ( θx A( θx B( θx C ( θx A( θx B ( θx C (- n p( x n p( x n p( x. ( θx A ( θx B ( θx C 4 ( θx A ( θx B ( θx C ( θ x (for D As in (- we obtain and 4 4 4 n p( x n p( x n p( x ( θx A( θx B( θx C ( θx D ( θx A( θx B( θx C ( θx D n p( x n p( x n p( x ( θ ( θ ( θ ( θ ( θ ( θ 6 x A x B x C x D x A x B n p( x n p( x n p( x n p( x n p( x n p( x. ( θx C ( θx D ( θx A ( θx B ( θx C ( θx D ( β and ξ ' ( ( N (- 0 N v( M' θ0 ' θ 0 ' (for The resuts are 7
n Pr( Z x n Pr( Z x m ( ( ( ( m ( m m m m ( X ( m ( m ( ( m( m n Pr( Z x n Pr( Z x ξ ( ( ( ( a ( x a a a a ( X x ( ( ( 8
n Pr( Z x n Pr( Z x ξ ( ( x ( ( ( X x ( ( a ( a a a a ( ( ( ( n Pr( Z x n Pr( Z x ξ ξ' ( ( xx' ( ( X xx' ( ( ( ( 9
n Pr( Z x n Pr( Z x ξ ξ ' xx' ( mm m m ( m m m m m ( m X xx' ( ( n Pr( Z x n Pr( Z x m ( ( x x x ( ( m aa a a a a a a a a b ( b b b b X x x xm ( aa a a ( a a ( a a a a a (- ( N (for ( 0 n Pr( Z x n Pr( Z x n Pr( Z x m ( ( m ( m m m m. 0
X m ( m ( m ( ( m ( m n Pr( Z x n Pr( Z x n Pr( Z x ξ ( ( a x ( a a a a X x ( ( ( ( ( n Pr( Z x n Pr( Z x n Pr( Z x ' ξ ξ ( a xx' ( a a a a X xx' ( ( n Pr( Z x n Pr( Z x n Pr( Z x m a x xxm ( a a X x xxm ( a a. a a a a (- ( ( ( N 0 0 (for
N ( θ ( θ ( θ ( θ n Pr( Z x n Pr( Z x n Pr( Z x n Pr( Z x = ( θ A ( θ B ( θ C ( θ D O N ( (see the expression of N A B C D Λ in. ( ( ( θ ( θ ( θ ( θ ( θ n Pr( Z x n Pr( Z x n Pr( Z x n Pr( Z x = ( θ A( θ B ( θ C ( θ D ( θ O N ( A B C D (see (- and the expression of Λ in. (. (-4 For ( ' ( ( N 0 0 (for ( ( 0 0 ( N see (-. Reca L n Pr( Z x θ θ ' remaining resuts are [{( L ( Λ }{( L ( Λ }] in AB AB CD CD then the n Pr( Z x n Pr( Z x [{( L AB ( Λ AB}{( L CD ( Λ CD}]. ( θ ( θ F We obtain n Pr( Z x n Pr( Z x ( Λ ( Λ m n m n ( (
( ( m ( m m ( m mn m m mn m m m m m m( n ( m n ( ( mn Λ Λ m m m n X mn 4 m( n ( mn ( m( n ( mn ( ( ( m n ( m( n ( m( n ( ( ( mn ( mn Λ Λ n Pr( Z x n Pr( Z x ( Λ ( Λ ξ m ξ x ( ( m m n ( ( m ( m m mm m( m m m m m
( m ( ( ( Λ Λ m m m m ξ m m m X x m ( m m m m m mm ( m m m m m m m ( m ( m m m m m mm ( ( m ( m m m m m m m ( m ( m m m ( m m mm ( ( m ( m m m ( Λ ( Λ m ( m m ξ n Pr( Z x n Pr( Z x ( Λ ( Λ ' ξ ξ xx' ξξ' ( ( ( ( ( mm m m ( m m m ( Λ ( Λ ξ ξ' m m m 4
X xx' ( ( ( ( ( ( ( ( ( Λ ( Λ ξ ξ' n Pr( Z x n Pr( Z x ( Λ ( ' ξ Λ ξ ξ ( ( xx' ( ( ( ( ( ( ( Λ ( Λ ξ ξ' X xx' ( ( ( ( ( ( ( Λ ( Λ ξ ξ' ξ' 5
n Pr( Z x n Pr( Z x ( Λ ( Λ ' ξ ξ ( ( xxx' ( ( ( mm m m ( m m m ( Λ ( Λ ξ ξ' m m m ξ ξ' X xxx' ( ( ( ( ( ( ( Λ ( ' Λ ξξ n Pr( Z x n Pr( Z x ( Λ ( Λ m n x x x x ( ( ( Λ ( Λ m n aa a a a a a a a a m n X x x xmxn aa a a ( a a a a a ( Λ ( Λ m n m n (-5 For ( ' ( ( N 0 0 (for ( N ( 0 see (-. The remaining resuts are 6
n Pr( Z x n Pr( Z x {( L AB ( Λ AB} ( θ C ( θ D n Pr( Z x n Pr( Z x X ( L AB ( Λ AB( Λ CD. ( θ C ( θ D We obtain the first term on the right-hand side of the ove equation as foows: n Pr( Z x n Pr( Z x n Pr( Z x m n ( ( ( ( m ( m n ( n m n m m n n X mn m( n ( mn ( ( m( n ( mn ( m( n ( ( mn ( m( n ( ( mn ( ( m( n 7
n Pr( Z x n Pr( Z x n Pr( Z x m ξ ( ( x ( ( m ( m a m m m a a X x m ( ( a a ( ( m ( ( ( m ( ( m ( ( ( m ( ( n Pr( Z x n Pr( Z x n Pr( Z x ξ ξ' ( ( xx' ( ( a ( a a a a 8
X xx' ( ( ( ( ( n Pr( Z x n Pr( Z x n Pr( Z x ξ m ( ( x ( ( ( m ( m m m m ( X x m ( m ( ( m ( ( ( m ( ( ( m ( 9
n Pr( Z x n Pr( Z x n Pr( Z x ξ ξ' ( ( xx' ( ( ( ( a a a a a X xx' ( ( ( ( ( ( ( ( ( ( n Pr( Z x n Pr( Z x n Pr( Z x ξ m ( ( x x xm ( ( a a a ( a a 40
X x x xm ( ( n Pr( Z x n Pr( Z x n Pr( Z x ' ξ ξ ( ( xx' ( ( aa a a a a a a a a ( ( ' ( ( X xx ( ( ( ( ( ( n Pr( Z x n Pr( Z x n Pr( Z x ξ ξ ' xx' x ( ( aa a a a a a a a a ( ( b b b b b 4
X xx' x ( ( ( ( ( ( n Pr( Z x n Pr( Z x n Pr( Z x m n x x x x ( ( m n aa a a a a a a a a b b b ( b b X x x xmxn ( aa a a ( a a ( a a a a a. (-6 For ( ' ( ( N 0 0 (for ( N ( 4 0 ( θ A( θ B ( θ C ( θ D ( θ N ( and N ( ( ( θ ( θ ( θ ( θ ( θ ( θ see (- an (.4. The remaining resuts are A B C D F Pr( Z x Pr( Z x N ( ( θ A( θ B( θ C ( θ D Pr( Z x Pr( Z x ( θ A ( θ B ( θ C ( θ D 4
in N ( ( θ A( θ B( θ C ( θ D ( θ ( θ F. Pr( Z x First we derive ( θ ( θ ( θ as foows: A B C n Pr( Z x ( m ( m ( ( ( m ( m ( ( m n Pr( Z x ( x ( ξ ( ( ( ( ( ( ( ( ( ( ( ( ( ( 4
n Pr( Z x ( xx' ( ξ ξ' ( ( ( { ( } ( { ( } ( ( ( n Pr( Z x {( ( x x xm a a a a } m a a Then the expectations are ( aa a a( a a ( a a a. a a ( n Pr( Z x n Pr( Z x ( m m n ( ( ( m ( m n ( n ( ( m n n n 44
= ( X mn 4 ( mn ( m ( m( n ( ( mn ( ( m( n m n ( ( ( ( ( m n ( ( n Pr( Z x n Pr( Z x ( x m m ξ ( ( 45
( m ( m a ( ( m ( a a a a = X x m ( ( ( ( ( m ( m ( ( m ( ( n Pr( Z x n Pr( Z x ( x ξ m ( ( ( ( ( ( ( m ( m ( ( ( m ( m m 46
= x ( X m ( ( ( ( ( m ( ( ( ( ( m ( ( ' ( n Pr( Z x n Pr( Z x ( xx ξ ξ' ( ( ( ( ( ( ( a ( ( ( ( a a ( a a = X xx' ( ( ( ( ( ( 47
( ( ( ( ( ( ( n Pr( Z x n Pr( Z x ( xx' ( ' ξξ ( ( ( { ( } ( ( ( { ( } ( ( = xx' ( X ( ( { ( } ( { ( } ( 48
( ( ( { ( } ( ( ( ( { ( } ( ' ( n Pr( Z x n Pr( Z x ( xx x ξ ξ ' ( ( ( { ( } ( ( ( { ( } ( a ( a a a a = X xx' x ( ( { ( } ( ( 49
( ( { ( } ( ( n Pr( Z x n Pr( Z x m n x x xm {( a a ( a a } a a n ( n ( aa a a( a a ( a a a n a a n n X x x xm {( n n ( n n } n n ( ( ( n n n n n n n n n n n n {( n n ( n n } n n ( n n nn ( n n n ( n n n n n n Pr( Z x n Pr( Z x m n 50
x x xm xn {( a a ( a a } a a b ( aa a a( a a ( a a a ( b b a a b b X x x xm xn {( a a ( a a } a a ( aa a a( a a ( a a ( a a. a a (-7 ( 4 ( 0 0 ( N N {( N } (for 4 n Pr( n Pr( x ( θ A ( θ B N N Z x Z N n Pr( Z x n Pr( Z x n Pr( Z x n Pr( Z x ( θ C ( θ D ( θ A ( θ B n Pr( Z x n Pr( Z x N( Λ ( Λ ( θ C ( θ D n Pr( Z x n Pr( Z x n Pr( Z x n Pr( Z x = ( θ A ( θ B ( θ C ( θ D ( Λ ( Λ AB CD (the fourth mutivariate cumuant of n Pr( Z x / ( θ s. The expectations required in the ove cumuants are n Pr( Z x n Pr( Z x n Pr( Z x n Pr( Z x m n A AB CD 5
( ( m ( m n ( n m n m m n n 4 4 X mn m( n ( mn 6 ( ( ( m n ( m( n ( mn n Pr( Z x n Pr( Z x n Pr( Z x n Pr( Z x m ξ ( ( x m ( m a m ( a a m m a a X x m ( ( ( m ( ( ( m ( ( m n Pr( Z x n Pr( Z x n Pr( Z x n Pr( Z x ξ ξ' ( ( a xx' ( a a a a 5
xx' ( ( X ( ( ( n Pr( Z x n Pr( Z x n Pr( Z x n Pr( Z x m n ( a x xm xn ( a a a a X x xm xn ( ( n Pr( Z x n Pr( Z x n Pr( Z x n Pr( Z x m n 4 a x x xm xn ( a a a a 4 X x x xm xn ( a a a a (-8 ( ( ( N 0 0 (for 4 The expectations are expressed as of two expectations (see (- (- and (-5. 0 ( where each ( is the product (-9 ( ( ( N 0 0 (for 4 5
The resuts are expressed as 5 ( where each ( is the product of three expectations (see (- and (-4. ( N (for 4 ( ( (-0 0 0 The resuts are simiar to those in (-9 (see (-6. (- (- ( θ A ( θ B ( θ (for C and The resuts are given by the Bartett identity (see (-. 4 ( θ A ( θ B ( θ C ( θ (for D and 4 The resuts are given by the Bartett identity (see (-. ( μ Σ β and ξ Some of the expectations of the og ieihood derivatives with respect to θx and θ e.g. N ( θx A ( θx B ( θ C ( θ are nonero D though the corresponding fourth cumuants are ero. The resuts are given as in (- (-4 (- and (-4.. The nonero partia derivatives of η with respect to θ. / We define S S( ξ Σ ( ξ' Σξ ( R then τ ( β ξ' μ S / and ρ (Diag Σ Σξ S.. First derivatives τ τ S S τ S τ τ S e S S τ S 54
where e is the vector of an appropriate dimension whose -th eement is and the remaining ones are 0. ρ ρ where Σξ S (Diag Σ ( ξ S ρs / / aa aa ba S S / (Diag Σ ( Σ S ρ S S S a b S ( Σξ S and is the matrix of an appropriate sie whose (a bth eement is and the remaining ones are 0.. Second derivatives τ S S τ τ S S S τ S cd ( cd cd cd τ S τ S es S S τ τ S τ S S S S τ S τ τ S τ τ S S S S τ S ( ρ 5/ / cd aa aaσξ S aa cd cd 4 ( cd 4 ( cd cd cd ( ρ S S ( acad adac ξ S S ρs 55
ρ ( Σ S (Diag Σ ( e e S / / aa aa a b b a ρ S ρ S S S S ρs ρ ρ S S S ρs ( where S cd ( ( cd cd S 4 S S S S S S S. ( ba S S. Third derivatives τ S S τ S τ S cd ( cd cd cd τ τ S τ S S S cd ef ( cd ef cd ef cd ef τ S S cd ef τ S S τ S τ S cd ( cd cd cd τ τ S τ S τ S S S S 56
τ τ S τ S S S S cd ( cd cd cd τ S S cd τ S S S S S τ S τ S cd cd cd τ τ S τ S τ S S S S τ S S τ S τ S m ( m m τ τ S τ S S S S ( τ S S τ S S S S S τ S τ S τ S S τ S τ S m ( m m τ τ S τ S S S m ( m m m τ S S m ρ 5 7/ 5/ cdef aa aaσξ S cd aa ef cd ef 8 ( cd ef 8 ρ S ρ S ( aeaf af ae ξ S S S cd ef ef cd ( 57
ρ S S cd ef ρ S 5/ / cd aa aa ( Σ S Σξ S aa cd 4 ( cd 4 S ( cd ( ac d a d c eas ( acad adac ξ S ρ S ρ S S ρ S S S S ( cd cd cd cd ρ S S S S S ρs ρs cd cd cd ρ ρ S ρ S S S S ( ρ S ρ S S S S S ρs ρs S ρ ρ S ρ S S S S ρs m ( m m m m where S cd ef ( ( ( 5 cd ef cde f S 8 S ( ( cd ( ba cds cd ( cd 4 cd S 4 4 S 58
S S ( ab b a S S S S S S S S S ( S S S S. m ( m m. Computation by Gaussian quadrature. Univariate case Stroud and Sechrest (966 Te Five pp.7-5 gave the foowing vaues of A and x : i i n exp( x f ( x dx Ai f ( xi i where n=(64(496(86. Let y x. Then ( y f ( y dy exp( y / f ( y dy n exp( x f ( x dx Ai f ( xi. The ove resut corresponds to Boc and Lieberman (970 quation (5. Boc and Lieberman (970 p.8 used n=64 where ony 40 vaues of 4 were empoyed since A i s for the remaining x i s are. Boc and Aitin (98 Te reported the resuts using n=0 and.. Bivariate case be i A i 0 Suppose that U ( X Y' N( μ Σ. Let the density of U at u ( x y' Σ ( x y exp ( ' ( / u μ Σ u μ x i 59
where Σ x yx xy y ( x y μ Σ. Define ( x0 ( x x that x x x and μ ( x y' ( U u ( x y and ( x ( x x exp x x. Then using the transformations ( x y g( x y dxdy ( x x x y y {( xy / x }( x x y y x y x / [ y {( xy / x}] y x [ y ( xy / x ( x y {( xy / x}] ( ( ( g( d d where i j x y x x x x y x y x y x x y x (note it foows y x g x y dxdy n ( A g( x d n x j x x x y x y x j x j n A A g( x x * i j x x i y x y x j. * y x ( / y xy x x y x i x x i 60
Part B. The partia derivatives of ULS with respect to ˆθ F F ˆ ˆ ˆ ˆ ULS ˆ Σ F ULS tr ( tr ( ˆ Σ Σ Σ ˆ Σ S ˆ ˆ ˆ Σ S ˆ ˆ ˆ ˆ i i i j i j i j F ˆ ULS Σ ( ˆ s ˆ i i F ˆ ˆ ˆ ULS tr ( ˆ Σ Σ Σ ˆ ˆ ˆ Σ S ˆ ˆ ˆ ˆ i j i j i ˆ ˆ j F ˆ ULS Σ ( ˆ ˆ ˆ ˆ i js i j 4 4 F ULS ˆ tr ( ˆ Σ ˆ ˆ ˆ ˆ Σ S ˆ ˆ ˆ ˆ i j i j 4 ˆ ˆ ˆ Σ Σ Σ Σˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ i j i j 4 F ˆ ULS Σ ( ˆ ˆ ˆ ˆ ˆ ˆ i js i j 5 5 F ULS ˆ tr ( ˆ Σ ˆ ˆ ˆ ˆ ˆ Σ S ˆ ˆ ˆ ˆ ˆ i j m i jm 5 ˆ 4 ˆ 0 ˆ Σ Σ Σ Σˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ i jm i j m 6
5 4 F ˆ ULS Σ ( ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ i js i j ( i j m... q; p a b. ˆF ULS Using the ove resuts with Lemma we have the partia derivatives of with respect to s as foows: Fˆ ˆ ˆ ULS tr ( ˆ Σ i Σ S ( ( ˆ ˆ Σ S s i s ˆ ˆ ˆ F ˆ ˆ ˆ ULS Σ Σ tr +( ˆ Σ i j Σ S s ˆ ˆ ˆ ˆ scd s i j i j scd ˆ ˆ ˆ ˆ tr ( ˆ Σ Σ S ( ˆ s i ( ac bd i i ˆ scd s i cd Fˆ ˆ ˆ ˆ ULS Σˆ Σˆ ˆ tr +( ˆ Σ i j Σ S ˆ ˆ ˆ ˆ ˆ ˆ sscd sef cd efi j s s s i j ˆ Σˆ Σˆ ˆ ˆ tr +( ˆ Σ i j Σ S ˆ ˆ ˆ ˆ s i j i j scd sef ˆ ˆ tr ( ˆ Σ i Σ S ˆ i s scd sef ˆ ˆ ˆ ˆ ˆ i j i ˆ ˆ s ˆ i j cd sef s i cd sef ( 4 4 Fˆ ˆ ˆ ˆ ˆ ULS tr Σ Σ ˆ ˆ ˆ ˆ Σ Σ s ˆ ˆ ˆ ˆ scd sef sgh i j i j 6
4 ˆ ˆ ˆ ˆ ˆ ˆ Σ i j +( Σ S ˆ ˆ ˆ ˆ i j s s s s ˆ ˆ ˆ 6 ˆ ˆ Σ Σ ˆ tr +( ˆ Σ i j Σ S ˆ ˆ ˆ ˆ ˆ ˆ s s s s i j i j cd ef gh cd ef gh Σˆ Σˆ ˆ ˆ ˆ tr +( ˆ Σ i j Σ S ˆ ˆ ˆ ˆ s i j i j scd sef sgh 4 ˆ ˆ ˆ 4 ˆ ii j tr ( ˆ Σ Σ S s s s s ˆ s s s s i cd ef gh cd ef gh 4 ˆ ˆ ˆ ˆ ˆ ˆ ˆ i j i j ( ˆ ˆ ˆ s s s ˆ ˆ s s s i j i j ˆ ˆ i ˆ s i cd sef s gh ( p a b p c d p e f p g h cd ef gh cd ef gh where instein s summation convention is to be used ony for subscripts i j and.. The chain rues gˆ gˆ Fˆ s Fˆ s ˆ ˆ gˆ gˆ F F gˆ Fˆ s s Fˆ s s Fˆ s s cd cd cd 6
gˆ gˆ Fˆ Fˆ Fˆ s s s Fˆ s s s cd ef cd ef gˆ Fˆ Fˆ gˆ Fˆ ˆ F s s s Fˆ s s s cd ef cd ef 4 4 gˆ gˆ Fˆ Fˆ Fˆ Fˆ s s s s Fˆ s s s s 4 cd ef gh cd ef gh 6 4 gˆ Fˆ Fˆ Fˆ gˆ Fˆ Fˆ ˆ F s s s s Fˆ s s s s cd ef gh cd ef gh 4 gˆ Fˆ Fˆ gˆ Fˆ ˆ F s s s s Fˆ s s s s ( p a b p cd ef gh cd ef gh c d p e f p g h. References Boc R. D. & Lieberman M. (970. Fitting a response mode for n dichotomousy scored items. Psychometria 5 79-97. Boc R. D. & Aitin M. (98. Margina maximum ieihood estimation of item parameters: Appication of an M agorithm. Psychometria 46 44-459. Ogasawara H. (0. Asymptotic expansions of the distributions of the poyseria correation coefficients. Behaviormetria 8 ( 5-68. Ogasawara H. (00. Asymptotic expansions of the nu distributions of discrepancy functions for genera covariance structures under nonnormaity. American Journa of Mathematica and Management Sciences 0 ( &4 85-4. Stroud A. H. & Secrest D. (966. Gaussian quadrature formuas. ngewood Ciffs NJ: Prentice-Ha. 64