Haruhiko Ogasawara. This article gives the first half of an expository supplement to Ogasawara (2015).

Transcription

1 Economic Review (Otaru University of Commerce, Vol.66, No. & 3, December, 5. Expository supplement I to the paper Asymptotic expansions for the estimators of Lagrange multipliers and associated parameters by the maximum likelihood and weighted score methods Haruhiko Ogasawara This article gives the first half of an expository supplement to Ogasawara (5.. Asymptotic expansions of the estimators with restrictions for model identification In this section, the estimator θ W identification is dealt with. Recall that in this case in (A.4, we have with restrictions for model η W. Setting W η l * n q * θw θ θ Λ / O ( n p / p O( n O ( n * q L Λ n ( θ W θ ( θ W θ θ O Op ( n p 3/ O ( n (S. 3 l ( θ θ ( θ ( θ W θ h Op ( n 9

2 4 l 3 ( θ θ 3 ( θ 3 W θ Op( n. 6 h 3 ( θ 3/ Op ( n From the above result, l * θw θ * * q Λ θ n Λ / * L Λ Λ ( θw θ O n ( * q * θ W θ Λ θ E ( J Λ θ θ ( θ O p O O p ( n p ( n 3/ ( n (3 T * h ( W Op ( n J E ( J Λ θ θ O (3 (3 * T ( W 3/ Op ( n 3/ O ( n (4 E T( J * 3 3 Λ ( W O 6 h θ θ 3 ( θ Recalling p (. p n O( n M L Λ, Equation (S. gives the following result: (S.

3 ( ( * θ θ Λ ( n Λ q W O( n θ / Op ( n Λ MΛ ( ( l θ l Op ( n (3 E T( J ( ( ( ( l Λ Λ h Λ ( θ θ Op ( n ( ( * ( ( ( ( n Λ MΛ q Λ MΛ MΛ 3/ O ( p n (3 E T( J ( ( ( ( l Λ M( Λ Λ h Λ ( θ θ n q θ * ( ( Λ Λ l θ 3/ Op ( n E ( J (3 T ( ( ( l ( Λ Λ h Λ (A ( θ θ (B n ( * ( ( Λ q Λ MΛ l θ 3/ Op ( n 3/ Op ( n l θ E ( J ( Λ Λ Λ ( p (3 T ( ( ( l h θ θ (B (A O ( n 3/ (S.3

4 Λ ( J E ( J Λ 3 i ( (3 (3 ( T (4 E T( J 3 ( ( ( 3 ( l Op ( Λ Λ n 6 h Λ 3 ( θ θ Λ l n Λ q O ( n ( ( i ( i ( * W p Λ ( i ( i i/ W l p In (S.3, l θ 3/ O ( n 3/ Op ( n O(, O ( n, i,, 3. Λ l Λ l Λ where ( ( ( ( ( W ML l θ Λ Λ Λ, l l / θ, ( ( ( ( W ML Λ l Λ l ( ( ( ( W ML (3 E T( J ( ( l ( ( ( l Λ MΛ ( Λ Λ h Λ θ ( θ θ ( ( l l ( ΛML ΛML v( M,, θ θ ( ( ( ( ( Λ Λ Λ with Λ and Λ where ML ML ML { ( / } q q q q and q ML p ( ML submatrices, respectively, being (S.4 (S.5

5 ( Λ ( Λ ( Λ ( ( ij ( ML ( ijk* ( ij i jk* * ( i j q; k,..., q, E ( J (3 T ( ( ( ( ( ML ( ij i j h ( θ ( Λ ( Λ Λ {( Λ ( Λ } ( i, j,..., q, Λ l Λ l Λ l (S.6 (3 (3 (3 (3 ( (W W ML W n, Λ l ( Λ Λ Λ Λ (3 (3 (3 (3 (33 (34 ML ML ML ML ML l l v( M, v( M, θ θ 3 (3 (3 l l vec{( J E T ( J },, θ θ ( (W ( ( l ΛW n l ( ΛW ΛW n v( M,, θ ( Λ ( Λ ( Λ ( Λ (3 ( ij ( k* l* ( ML ( ijk* l* m i jk* l* m ( ij ( k* l* * * ( i j q; k l q; m,..., q, 3

6 ij ( Λ ( i (3 ( ML ( ijk* l* Λ ( ij (3 E T( J ( ( ( ( ( Λ Λ h {( Λ k* ( Λ l* } ( θ (3 E T ( J ( ( ( ( ij ( ( Λ Λ h ( Λ k* ( Λ i ( Λ jl* ( ij ( θ * * ( i j q; k, l,..., q, ( Λ ( Λ ( Λ ( Λ (33 ( ( ( ML ( ijk* l* m i jl* k* m * * ( i, j, k, l, m,..., q, E ( J ( Λ ( Λ Λ ( Λ (3 T (34 ( ( ( ML ( ijk* h ( θ (A * ( i, j, k,..., q, (3 E T ( J ( ( ( ( ( Λ Λ h {( Λ j ( Λ k} (B ( θ (B (A (4 E T( J ( ( 3 ( ( ( ( Λ Λ {( i ( j ( k* } 6 h Λ Λ Λ 3 ( θ i j 4

7 ( Λ ( Λ ( Λ q ( ( ij ( * W ( ij i j ( ij ( i j q, q ( Λ Λ ( Λ * ( ( ( W i θ E ( J (3 T ( ( ( ( * h i ( θ ( Λ Λ {( Λ ( Λ q } ( i,..., q. Equation (S.3 is alternatively written as i 3 ( i ( i ( (w ( * W ML W n n Op n i θ θ Λ l Λ l Λ q ( θ θ Λ n l n Λ q O ( n. ( (W ( * ML W p (S.7. Properties of augmented matrices Define column-wise q blocks of a matrix A as A (,..., ( j j q A ( A A A. Similarly, define row-wise p blocks of A as ( ( ( q A (,..., ( i i p i.e., A ( A( A( A( p. Let B ( B( B( where B( and B( are b q and b r submatrices, respectively; i.e.,, I( q O q r, q r or q r. Post-multiply B by G C I. Then, BG ( r becomes ( B( B( C: B( with the colon being used to show separation for clarity. Matrix B is restored from BG by subtracting B( C from the first column-wise block of BG, which is given by * BGG B, G * I( q O C I. Since ( r * G G. That is, we have elementary results: 5

8 Lemma S. I( q C I( q C O I O I. ( r ( r I( q O I( q O C I C I and ( r ( r B D Let C O be a non-singular ( q r ( q r matrix, where the q q diagonal block B may be asymmetric and singular. Lemma S. The first column-wise and row-wise blocks of 6 B D C O B DC D equal to those of C O, respectively. Equivalently, the two matrices are different only in their second diagonal blocks. B DI( q O B DC D Proof. Since C O C I( r C O, B D I( q O B DC D C O C I( r C O. Using Lemma S, I( q O B D B DC D ( r C I C O C O. Noting that pre-multiplication of a I( q O matrix by C I does not change the first row-wise brock of the ( r multiplied matrix, we find that the first row-wise blocks of the inverted matrices are equal. Similarly, from I( q D B D B DC D ( r O I C O C O we have B D B DC D I( q D C O C O O I. Using Lemma S, ( r are

9 B D I( q D B DC D C O O I( r C O, which shows that the first column-wise blocks of the inverted matrices are equal. Q.E.D. B D When B is non-singular and when q r, C O obtained by the formula of the inverse of a partitioned square matrix is straightforwardly E F E E FJGE E FJ G H, where J ( H GE F with JGE J the assumption of the existence of the inverses, which gives B D B B D( C B D C B B D( C B D ( ( C O. C B D C B C B D When B is possibly singular, and B D C O with q r and B DC are non-singular, we have Lemma S3. Let A B DC. Then, B D A A D( C A D C A A D( C A D C O ( ( r ( C A D C A I C A D. Proof. Using the result in the proof of Lemma S, and the formula of the inverse of a partitioned matrix, B D I( q O B DC D C O C I( r C O I( q O A A D( C A D C A A D( C A D ( r ( ( C I C A D C A C A D A A D( C A D C A A D( C A D. Q.E.D. ( C A D C A I( r ( C A D In the case of the augmented information matrix per observation q > r, we have * I with 7

10 ( ( * I H I C I I Theorem S. Let I ( ( H O C O, where I I I may be singular; A I CC and C A C are non-singular. Then, I A A C( C A C C A A C( C A C C A C C A I C A C. Proof. Use Lemma S3 with B I and D C. Q.E.D. / / It is known that the asymptotic covariance matrix of ( n θ, n η * ( ( r ( * I O * under correct model specification is given by I I O O, whose alternative expression (see Silvey, 959, Lemma 6; Jennrich, 974; Lee, 979 is given by the following: Corollary S. When I in the augmented matrix may be singular, ( * I O I * O I ( O O O I I. Proof. * I O * A A C( C A C C A I I ( ( A CC O O C A C C A I O A A C( C A C C A O ( I ( A A C( C A C C A A C( C A C O C A C I O Q.E.D. ( ( r * When I is non-singular, since I is obtained by the formula of the inverse of a partitioned matrix as * ( ( I C I I C C I C C I I C C I C I C O ( (, it C I C C I C I C 8 W W

11 follows that * * ( I O I I C C I C C I O I I O O ( O C I C ( I O, ( O I which is also known (Aitchison & Silvey, 958, Theorem. ( ( ( The result I II I in Corollary S shows that I is a ( generalized inverse of I. However, the reverse is not true (Jennrich, 974, which is shown as follows. I I I ( A CC{ A A C( C A C C A }( A CC ( A CC CC A CC ( A CC A C( C A C C A ( A CC A CC CC A CC { C( C A C C CC CC A CC} A C C A C C ( I C I C I CC C C I [ ( r { ( } ], which indicates that I ( I is not a generalized inverse of I. I Noting that Theorem S holds when 9 is non-singular as well as when is singular, we have Corollary S. When I is non-singular and A I CC, A A C( C A C C A A C( C A C ( C A C C A I ( C A C ( r I I C( C I C C I I C( C I C ( C I C C I ( C I C.

12 Proof. An alternative direct proof of Corollary S is given as follows. First, we derive the equality of the second diagonal blocks of the above equation. Since C A C C( I CC C C{ I I C( C I C I C I } C ( r C I C C I C( C I C I ( C I C I I ( r ( r ( r C I C( C I C I, ( r I ( C A C I ( C I C I ( C I C ( r ( r ( r ( C I C. The equality of the lower-left (or upper-right blocks is given as follows. ( C A C C A { I ( C I C } C{ I I C( C I C I C I } ( r ( r { I ( C I C }[ C I { I ( C I C I } C I ] ( r ( r ( r { I ( C I C }( C I C I C I ( r ( r { I ( C I C }[ ( C I C ( r ( C I C {( C I C I } ( C I C ] C I ( r { I ( C I C }{( C I C I } ( C I C C I ( r ( r ( C I C C I. The equality of the first diagonal blocks is given as follows. A A C( C A C C A A A C( C I C C I I I C( C I C I C I ( r { I I C( C I C I C I } C( C I C C I ( r I I C( C I C C I. Q.E.D. A different expression of the result of Lemma S3 when B is non-singular is given from the result before Lemma S3 as

13 A B DC as in Lemma S3, Lemma S4. When B is non-singular and B D A A D( C A D C A A D( C A D C O ( ( r ( C A D C A I C A D B B D( C B D C B B D( C B D. ( ( C B D C B C B D Let C and D in Lemmas S, S3 and S4 be partitioned as C ( C( C( and D ( D( D(, where C( and D( are q s submatrices, and C( and D( are qt submatrices with s t r after possible simultaneous permutations of the columns of C and D. Assume that B D C ( ( is non-singular, where B may or may not be singular. The quantity t ( t r is not necessarily the minimum value to yield the non-singular B D C ( ( when B is singular. Lemma S5. The first and second row-wise blocks of B D C O B D ( C( D corresponding to the first q rows and the following s C O rows are equal. Similarly, the first and second column-wise blocks of the matrices corresponding to the first q columns and the following s columns are equal. Proof. From the identity, I( q O O B D B D( C( D O I ( s O C O C O C( O I( t I ( q O O B D ( C ( D O I( s O B D we have C O C O C( O I( t Similarly, from the identity, and

14 I( q O D ( B D B D( C( D O I( s O C O C O, O O I( t using Lemma S we have I ( q O D ( B D B D( C( D O I( s O C O C O O O I( t The above results show the required equalities. Q.E.D. Define A B D C ( ( (. Then, using Lemma S5, we have Theorem S. The three submatrices other than the second diagonal block on the right-hand side of the identity B D( C( D C O A( A( D( C A( D C A( D( C A( D ( ( ( ( ( C A D C A C A D are unchanged irrespective of the choice of C( and D(. Theorem S3. Theorem S holds even if B D C ( ( is replaced by B D EC ( (, where E is a non-singular t t matrix. Proof. From the identity I( q O O B D B D( EC( D O I( s O C O C O EC( O I( t I ( q O O B D( EC( D O I( s O B D C O C O EC( O I( t, we obtain. Similarly,

15 I( q O D ( E B D B D( EC( D O I( s O C O C O O O I( t I( q O D ( E B D( EC( D O I( s O gives B D C O C O. These two results O O I( t B D B D ( EC( D show that C O and are the same except C O their second diagonal blocks. Q.E.D. Crowder s (984, Lemma 6 result for the augmented matrix derived in a different way is a special case of Theorem S3 when B I and D C H. Note that the arbitrariness of E corresponds to the arbitrary expression of the restriction Eh = with E being non-singular. Using Lemmas S, S5 and Theorem S, we have Corollary S3. When B is possibly singular, I ( q O O B D B D( C( D O I( s O C O C O C ( O I( t A( A( D( C A( D C A( D( C A( D. O O ( C A( D C A( ( C A( D O I(t Equating the results of Lemma S3 and Corollary S3, we find that O O ( C A( D I( r ( C A D O I or equivalently ( t I( s O ( C A D ( C A( D. When B is non-singular. O O 3

16 O O ( C A( D ( C B D O I. ( t 3. Applications of the general formulas to Example. (Example. 3. Non-studentized estimators n n l x n x n n, θ ( ( i i i i m n m n n,, ( ( n m n ( l ( { ( } n, θ θ n m n ( ( { ( } n l ( E n θ θ n ( c ( c ( I ( c n / n O(, c n / n O(, 4

17 l l M E θ θ θ θ m n ( { ( } n, m n ( { ( } c l l ( E n n I, θ θ c ( l c E ( M n (,, θ { ( } l c E ( M n, (, θ { ( } l l E ( M E ( M, θ θ Λ I * I H H. Since, define ( ( c ( c and I θ θ. Then, using c ( c, 5

18 Λ I I (, I (, I sym. * (, I (, I (, I c c ( cc ( c, c sym. c c, cc cc ( c, c ( c c c ( c c c cc ( c, c ( c ( c, cc ( c, c ( * k k k k q, k, ( ( k( (,. ( ( ( ( ( ( l ( l W ML θ ( c, c θ Λ l Λ l Λ, ( ( Λ Λ ( ( Λ ( Λ ( ( Λ, l l / θ where W ML. 6

19 n n Λ * k where ( J ( ( ( q ( n Λ ( c, c ( k(, ( c c /{ ( } Λ ( q * ( k. For nonzero elements of (3 (, (3 J, n( ( n ( m n 3 { ( } { ( } { ( } n( ( n ( m n, 3 { ( } { ( } { ( } (3 E {( J (, } c, { ( } { J E ( J } (3 (3 (, Similarly, E ( n ( m n. 3 { ( } { ( } {( J } c, (3 (, { ( } { J E ( J } (3 (3 (, ( n ( m n. 3 { ( } { ( } 7

20 Nonzero elements of E E (4 J are (4 3( 3 (, c 3 { ( } { ( } {( J } ( 6 c, 3 { ( } { ( } {( } 6. (4 ( J (, c 3 { ( } { ( } ( ( ( ( ( ( l l ΛW l ΛMLl ( ΛML ΛML v( M, θ θ, where ( ( ij ( ( ΛML ( ijk* ( Λ ( i ( ij Λ jk* ( ij ( ij ( c, c i { ( } { ( } ( ij ( ( ( c, c i( ij / ij * ( i j ; k,, 8

21 (3 ( * E ( ( ( ( ML J Λ ( ij Λ {( Λ i ( Λ j} O ( (3 ( ( Λ E ( J {( Λ i ( Λ j} ( (3 E ( J { ( } ( c, c ( ( c, c ( i, j,..., q. c c ( ( ( Λ l Λ l Λ l (3 (3 (3 (3 ( (W W ML W n, Λ l (3 (3 ML l l ( Λ Λ Λ Λ v( M, v( M, (3 (3 (33 (34 ML ML ML ML θ θ 3 (3 (3 l l vec(( J E T ( J,, θ θ 9

22 ( (W ( ( l ΛW n l ( ΛW ΛW n v( M,, θ ( Λ ( Λ ( Λ ( Λ (3 ( ij ( k* l* ( ML ( ijk* l* m i jk* l* m ij k l ( ( ( ij k* l* { ( } ( ij ( k* l* ( c 4, c i 3 { ( } ( ij ( * * k l ( c (, c i{( ij / }{ ( } ij ( ( * * * * ( i j ; k l ; m,, ( Λ (3 ML ( ijk* l* ( ij ( ij ( ij ( (3 ( ( ( Λ i Λ E ( J {( Λ k* ( Λ l* } ( (3 ( ( ij ( Λ E ( J ( Λ k* ( Λ i ( Λ jl* ( ij ( ij ( c ( Λ i ( Λ j ( c ( c Λ ( ( ( ij c j 3

23 ( ij ( ij ( c, c i ( ( ( ( c ( ( ( ij c ( c, c 3( { ( } ( ij ij ( c (, c i ( ( ij * * ( i j ; k, l,, ( Λ ( Λ ( Λ ( Λ (33 ( ( ( ML ( ijk* l* m i jl* k* m ( { ( } ( c, c ( { ( } ( c, c i * * ( i, j, k, l, m,. ( Λ (34 ML ( ijk* i ( (3 ( ( (3 ( ( Λ E ( J ( E ( {( } Λ i Λ J Λ j Λ k ( (4 ( ( ( Λ E ( J {( Λ i ( Λ j Λ k* }, 6 where the first term is 3

24 ( (3 ( ( c c Λ E ( J ( Λ Λ ( i ( (3 Λ J E ( ( ( ( c Λ 4 ( ( c ( ( and the second term is ( c Λ 6 {( ( } 6 c [ ( ( { ( } ]. Then, (34 ( ΛML ( ijk* ( ( { ( } ( ( ij ( * ( ΛML ( ij ( Λ i ( Λ q j ( ij ( ij ( Λ i { k( } ( ij ( ij k ( ij ( c, c i ( ( ( ( ij k( ( c, c i ( ij ( ij ( i j,. 3

25 k( ( (,, where q ( Λ (, and consequently, * ( c c ( * ( Λ q k is used. q ( Λ Λ E ( J {( Λ ( Λ q } Λ ( Λ * ( ( (3 ( ( * ( ( ML i i i θ J k E ( ( ( ( (3 Λ ( Λ I ( Λ Λ ( ( k ( ( { ( } ( c 4 k( c ( ( ( k ( ( { ( } Λ ( 4 k( ( { ( } ( k( ( c c { 4 ( ( } k ( i,. ( ( c c ( i 3. Studentized estimators 3.. t W ( tw and t W Using and rather than for differentiation, 33

26 I Ι H { E ( Λ } H * * Ι Ι (, Ι (, Ι Sym. where (, Ι (, Ι (, Ι I c ( ( c c ( ( c Then, noting I ( / c ( / c ( ( (, Ι c c and, it follows that * A b I b d, where ( c ( ( A ( c c c ( c ( (,, ( c c c ( ( ( ( b,, c c c c., 34

27 ( ( and d c c. * Since direct differentiation of I with respect to k is tedious, we use the * * * * formula of I / I ( I / I, where I I k* k* c ( Ι H c ( c ( c cc ( c, c ( * H * looks simple. I ( i I I ( * ( * * ( 3/ j i j though c ( { ( } { ( } { ( } j 3/ c j ( { ( } ( j,, 3/ after evaluation using I c ( i I I ( ( { ( } * ( * * ( 3/ j 3/ j i j ( j,. ( ( i i. Note that That is, 35

28 * * * I * * I * I I I I c j ( ( j, j. j / / In tw n ( i W ( W, î W * I with θ θ replaced by W and becomes W ( W / / t W n { W ( W } ( W, where note that t W / n element of is the first or second diagonal. Then, is defined using / rather than n W. That is, W and t W reduce to those of a single group proportion with size n and pseudocounts 4k in total. ( ( i ( jk* I I I I I I I I 4 * * * * * * * * 3/ { ( } j k* j k* 3 I I I I I I 8 * * * * * * 5/ { ( } j k* c j ( ( jk* c jck* ( } ( ( 3 5/ c jck* ( { ( } 8 * ( j, k,, where 3/ c( * { ( } * I I c (, { ( } and non-zero second derivatives are 36

29 ( c 3 * { ( } { ( } I (, ( * I c 3 { ( } { ( } (. ( ( i i (, ( i ( jk* ( i ( jk*. Incidentally, ( ( ( i ( j, k* ( i ( j, k* j k* j k* c c ( ( ( c c( c c { ( } ( ( 3 ( 5/ c c ( c c ( { ( } 8 3/ 3 5/ { ( } ( { ( } 8 ( i, which is a scalar in the case of a single group. 3/ 3.. t W where / / ( i W / cc ( i about (, * cc avar( n W n ( I n (. Expand 37

30 ( I i ( i ( i ( * / / * * 3/ W I I θw θ j j I I I * * * * * * * * 3/ I I I I I ( i j k* 4 j k* j k* * 3 * I * * * I * 5/ I I I I ( i 8 j k* 3/ ( θw θ j( θw θ k* Op( n ( i i ( θ θ i ( θ θ O ( n, where I I / ( ( 3/ W W p I c ( * * * I c c cc { ( } { ( } I * * * I c c { ( } consequently, * I ( i j I I ( i j ( * * 3/ where ( cc cc c j { ( } ( j / ( cc c { ( } I I / ( c c, ( j,, 3/ * * * I j c j { ( }., j 38

31 ( jk* ( jk* 3 c j { ( } { ( } ( i ( c c ( * ( cc ( jk ( cc 3 { ( } ( c c ( cc 4 8 c jck* { ( } ( ( jk* ( c c ( { ( } 5/ ( c c ( { ( } { ( } / jk* / c j 3/ / 3/ 3 ( c c ( { ( }. 3/ 3/ 8 c jck* / / Note that while tw n ( i W ( W, / / t W n ( iw ( n W, where nw cwc W( pw pw / n W Op ( n n (. Alternatively, W W 3/ t n ( i. / / W W W 4. Results using special properties of Example. (Example. 4. Partial derivatives of with respect to p and p Note that cc ( p p c p c p and n (, then (, p p n p θ c c ( ( c, c ( { ( p } c c (,, ( p θ 39

32 n c c ( ( p { ( } c c cc p θ c ( c p p p p ( 3 { ( } c { ( } c c c c c ( c c { ( } c c ( ( c, c c, c c, c, { ( n } ( 3 c 3 c pθ 3 ( p { ( } { ( } c c c c c c c c c c 3 { ( } { ( } {( c, c c, c c c ( p θ, c c c, c c c, c c c, c c, c ( c, c, c c, c c, c c, c c, c, c } ( cc 3 ( 3 c, c cc, c cc, { ( } { ( } where c c c c c, c c c, c c c, c c c,3c ( c, cc, cc, c (, ( c, c, c c, c c, c c, c c, c, c. 4. Asymptotic cumulants of and by ML 4

33 c p c p is a usual sample proportion of size n, Since (, ( n (, ( n ( (, 3 3 4( n ( { 6 ( }. On the other hand, since cc ( p p cc ( p p n ( ( c p c p( c p c p, we use ( p, ( p n ( vec diag( c, c, ( p n ( ( ( c,, c, 3 (6 ( p n ( { 6 ( }( c,, c (4 ( n cov( p ( diag( c, c. n ( n E {( } O( n p p ( p n n ( vec diag( c, c O( n ( p c c c c ( c c vec diag(, n c c ( O n n O n ( c c ( ( O( n ( ( (. 4

34 n p ( n n ( diag( c, c n n n n 3( p, p ( p, p p ( p 4 ( p ( p 3 n 3 n 3 n ( p, p O( n 3 3 p ( p (, (, n cc ( diag( c, c cc ( ( n p (, cc ( c c n cc ( { ( } ( ( ( c,, c (6 n n ( p { n cov( p } n ( p c c 3 n 3 n n [ n cov( p vec{ n cov( p }] O( n 3 p ( p c c n n c c ( ( ( cc { ( } cc 4 c c c c c c { ( } n ( ( { ( } (,,, diag,,, ( c, c c, c c, c c cc cc c ( cc c n (, ( c,,, c ( c ( { 3 ( } { ( } { ( } ( 3 c, c c c, c c c, c c c, c c c, c c c, c c c,3 c O( n 3 4

35 c c n n c c n ( ( { ( } n ( cc ( ( c c 4 4 { ( } cc ( ( cc ( { ( } 3 (,( 3 c c, cc 3 O( n cc ( ( { ( } ( n {4( cc cc ( c c } { ( } ( 3 ( { ( } cc cc ( 3 n n c c n ( c c ( c c c c O( n n n O( n ( ( ( n n O n 3 (, where 4( cc cc ( c c cc and are used. 3 n 3 n n n 3 ( ( p p n (3 3 n n ( p p 3 n 3 n vec ncov( p O( n ( p c c c c / ( c c {vec ncov( p } 43

36 ( c c n (, ( ( ( c,, c (6 { ( } n 3 n n n 3 ncov( p ncov( p O( n p p p p c c ( c c ( c c n ( 3 ( 3 n { ( } { ( } c c c c c 3 (, O( n c c c c c 3 cc ( c c ( cc n ( n 3 ( { ( } { ( } c c c c 3 ( c, c O( n c c c c c c ( c c ( c c n ( 3 ( 3 n { ( } { ( } c c 3 c c O( n cc n c c ( c c O( n n 3 { ( } O( n,

37 4 3 n 3 4 n n n 4 ( ( p p 3 ( 3 n n n {vec ncov( n 3( } p p p ( p n p ( p 3 p ( p n n n n 3 (5 {vec ncov( p } 6 n O( n, where the first term is n 4 3 n 3 p n ( p ( c c n (, ( { 6 ( }( c,, c (4 { ( } 6 ( n ( c c c c { ( } n cc ( 3 cc 6, { ( } ( the second term is 45

38 3 ( 3 n n n {vec ncov( n 3( } p p p ( p ( c c n ( {(, ( c, c c, c c, c } { ( } ( c c c (6 c { ( } ( (,,, (,, 4 3 ( cc 3 n ( 3 {(, ( c, c c, c c, c } { ( } ( ( c,,, c ( c, (6, c and the third term is n p ( p 3 p ( p n n n n 3 (5 {vec n cov( p } 3 3 ( c c n ( {(, ( c, c c, c c, c } Then, { ( } 4 ( cc ( { ( } { ( } { ( } {(, ( 3 c, c c c, c c c, c c c, c c c, 3 (5 3 3 cc c, cc c,3 c }{ ( } ( c,,, c. 46

39 c c ( 3 c c ( n n 6 { ( } ( 3 4 (A ( c c 4 3 ( 3 {(, ( c, c c, c c, c} { ( } ( ( c,,, c ( c, (6, c 3 ( ( c c {(, ( c, c c, c c, c } 4 3 { ( } (B 4 ( 3 { ( } { ( } 3 {(, ( 3 c, c c c, c c c, c c c, c c c, 3 (5 3 cc c, cc c,3 c } 4 ( c,,, c 6 O( n n O n (, (B 4.3 Asymptotic cumulants of t and t by maximum likelihood / n ( t / { ( }, where is the MLE. The asymptotic cumulants of t reduce to those of the studentized sample proportion (see e.g., Ogasawara,, p.: (A 47

40 p, q, ( t 3 3 / / ( pq 3/ / ( t /3 ( t n ( p O( n n O( n, 7 ( t ( t n ( p ( pq 3 O( n n O( n 4 (, ( t n ( pq ( p O( n n O( n / / 3/ / ( t / ( t n {( pq 9( p ( pq 6} O( n n O( n, 3/ ( t 4 4 ( t / / n acov(, ( p ( pq ( pq, 4 n acov(, ( p ( pq 4( pq. t ( t / / 3 ( ( / { ( } / / n n n cc p p / / ( i [ cc / { ( }] n ( c c ( p p p p / / / / { ( } { ( / ( ncc } cc ( p p where n cc ( is the MLE and i (. ( Noting that ( ncc n n and var( p p ( n n p p., 3,, we find that t is the studentized 48

41 ( p p ( c p c p (,, ( c, c, p p p t / / (, p p ( n cc ( ( / 3/,, c c { ( } { ( p } t c c n ( c c p c c / / 3/ ( { ( } 3 ( ( c p p, 3/ 5/ { ( } 4 { ( } c 3 t / / 3 ( n ( c 3 c 3/ 5/ ( p { ( } 4 { ( } c c c c. c c c c 3/ ( t E {( } O( n p p c c / / / 4{ ( } 3/ ( / t ( p t 3/ n ( vec diag( c, c O( n ( p n ( c c ( c c O n n / ( cc ( 3/ / ( ( O n { ( } O( n (, 3/ ( t vec diag( c, c 49

42 t t ( t n ( diag( c, c p p t t t t p ( p 4 ( p ( p 3( p, p ( p, p 3 t t 3 ( t ( p, p n ( 3 O( n, 3 p ( p where the first term is c c c ( (, c ( the second term is c c ( n { ( } c c c c ( ( ( ( c c ( ( n ( c c n, the third term is n t { n cov( p } ( p ( p, ( c, (6, c c c ( n { ( } ( c, c c, c c, c 3 8{ ( } diag( c, c c, c c, c ( c, c c, c c, c cc ( ( c c n ( cc n ( 4 ( and the fourth term is t 5

43 n c c 3 ( (,[ n cov( p vec{ n cov( p }] / 3/ 5/ { ( } { ( } 4 { ( } c c c c c c c c 3 ( c n cc (, ( c 4 ( c,,, c ( 3 c, c c c, c c c, c c c, c c c, c c c, c c c,3 c 3 ( n cc (,( 3 c c, cc 3 4 ( 3 ( n cc ( c c 4 ( n 3 (. 4 ( Consequently, n O( n ( t ( t c c 3 ( ( t n O( n 4 4 ( n O( n (. 3 (3 t 3 t t 3( t 3( p {vec cov( } p p ( p p n 3 O( n, / ( t 3/ where the first term is 5

44 ( c c n ( ( (, ( c,, c 3/ / 3 3/ (6 { ( } ( c c ( n ( c c n 3/ / / { ( } ( c c ( / / { cc ( } the sum of the second and third (actually terms is 3 cov( p cov( p p pp p t t t ( c c 3 n { ( } 3/ / 3/ ( { ( } c c c c c c (, (, (, c c 3/ / 3 ( cc ( / (, c c c c { ( } c c c c n c c, where the quadratic form on the left-hand side of the last equation is. Then, ( c c ( ( t n O( n / 3/ 3 / { cc ( } n O( n. / ( t 3/ 3 5

45 p p ( p 4 3 t t t 4( t 4( p ( {vec cov( p ( p } 3 t t t t ( 3 p p p ( p (5 3 ( t 3 {vec cov( p } n 6 O n where the first term is (, ( c c n ( { 6 ( }(, ( c,, c (4 { ( } c c n 6. c c ( Since becomes n 3 3 c c c c 3 cc ( c c 3 c c c c c c c c, the first term 3 6. c c ( The second term is ( c c n { ( } ( 3/ 3/ { ( } { ( } ( 3 c c c c c c c c c (6 c {(, (,,, } (,,, (,, and the third term is 3 53

46 3 ( c c ( n {(, ( c, c c, c c, c } 4 { ( } 4 ( c c ( 3 3/ 3/ 5/ 3 { ( } { ( } 4 { ( } (, ( 3 c, c c c, c c c, c c c, c c c,c c Then, (B 3 (5 3 3 cc c,3 c { ( } ( c,,, c. 4( t n 3 6 cc ( (A 3 ( ( ( ( c c {(, ( c, c c, c c, c } ( c,,, c ( c, (6, c 3 ( cc ( {(, ( c, c c, c c, c } 8 ( 3 ( ( cc 3 4 ( c (, ( 3 c, c c c, c c c, c c c, c c c, c c c, n 3 (5 3 ( t cc c,3 c ( c,,, c 6 O( n O( n. ( t 4 ( t nacov( n, (B (recall that, ( t (A, 54

47 ( / t p p nacov( n, 3 ( cc ( c c n acov, / ( { ( } ( c c ( ( { ( } { ( } / ( c c / 3/ c c c ( (, c. 4.4 Asymptotic cumulants of W and W by the weighted score method m k n k n 4k n k W n 4k n 4k n 4k n 4k n 4k n k O ( n n k( O ( n. p p In the following, only the asymptotic cumulants different from those by ML are shown. ( n n k( O( n W n k( O( n n W O( n (recall that, ( n n ( 8 k O( n W 3 n n W O( n ( W ; W. nwcwcw ( p W pw W, where W ( W n k n k nw n 4 k, cw, cw, n 4k n 4k m k m k p W, pw, n k n k 55

48 c n k W c c n k( c O( n, n 4k c c n k( c O( n, W p n k p n c k p p n c k( p O ( n, W p n k n c k p p n c k( p O ( n, W p p p p p n { ( k c c c p c p } O ( n, W W { ( } { ( } { ( } ( ( O ( n W W W p { ( } n { ( } k( Op( n. From the above results, n n ( n 4 k{ c n k( c }{ c n k( c } W [ p p n k{ c c ( c p c p }] k( n O ( p n ( { ( } cc ( p p ( k( n 4k kc ( c kc ( c ( cc ( p p cc k{ c c ( c p c p} Op( n ( ( 4 n n k k k( c c ( k c c c p c p O ( n Then, { ( } p(. n p 56

49 ( c c( ( n W n n k O( n ( ( c c ( n k O( n ( n O( n W (recall that and, ( ( n n n 4k cc ( W where ( c c ( n k ( (, c c c { ( } c p ( c p c p c n 3 4 k O( n c p p ( n n k 4 cc ( (A (B cc ( n n O( n (, 3 W W O n (B (A 3 ( c n ( c, c c p and ( c p c p c n c c c ( c, c c c p p ( c c cc cc c c ( ( n cc p ( are used with. 57

50 References Aitchison, J., & Silvey, S. D. (958. Maximum-likelihood estimation of parameters subject to restraints. The Annals of Mathematical Statistics, 9, Crowder, M. (984. On constrained maximum likelihood estimation with non-i.i.d. observations. Annals of the Institute of Statistical Mathematics, 36, A, Jennrich, R. I. (974. Simplified formulae for standard errors in maximum likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 7, -3. Lee, S.-Y. (979. Constrained estimation in covariance structure analysis. Biometrika, 66, Ogasawara, H. (. Cornish-Fisher expansions using sample cumulants and monotonic transformations. Journal of Multivariate Analysis, 3, -8. Ogasawara, H. (5. Asymptotic expansions for the estimators of Lagrange multipliers and associated parameters by the maximum likelihood and weighted score methods. Submitted for publication. Silvey, S. D. (959. The Lagrangian multiplier test. The Annals of Mathematical Statistics, 3,