Supplement to Ogasawara s papers on the ADF pivotal statistic, mean and covariance structure analysis, and maximal reliability. Haruhiko Ogasawara

Transcription

1 Economc Revew (Otaru Unversty of Commerce, Vol.60, o., -44, July, 009. Sulement to Ogasawara s aers on the ADF votal statstc, mean and covarance structure analyss, and maxmal rellty Haruhko Ogasawara In Sectons through of ths note, sulements to Ogasawara (008, 009a, 009b are gven, resectvely.. Sulement to the aer: Some roertes of the votal statstc based on the asymtotcally dstrbuton-free theory n structural equaton modelng Snce the roof of Theorem n Ogasawara (008 was somewhat ndrect, another roof of the theorem more drectly derved s gven n ths secton. acov ( θα, = acov ( θα, We show only the essental result T T T requred for the roof of Theorem. From (., n acov ( s, m = T cdef cdef cdef c def 4 5 = = Usng the ove result, cd ef cd ef ac bd ef θ α nacov ( θα, = Ω θ Ω T 4 ' ' 4 θ α θ α = Ω nacov ( sm, ' ' T T 4 ' 4 θ α = ( ag bh ah bg a b g h gh θ α acbdef a b c d e f cdef. α (A.8 (A.9

2 θ = ( ag bh ah bg a b g h α θ ( θ T k l l k kl gh k l gh kl θ θ θ abkl a b k l kl θ α T θ = ΩT ( ag bh ah bg ' gh,,,,, kl,, = θ ( k l l k kl θ θ θ θ,,, kl,, = kl where s a b ; a b and ba ( b a gh kl c d e f, s a b kl ; and denotes that c d e f are temorarly treated as dfferent varles n dfferentaton. The second term on the rght-hand sde of the last equaton of (A.9 becomes θ ( ag bh ah bg gh,,, = θ θ θ θ l l, = g hl k, = g kh θ θ θ θ k k, = g kh, l= h gl θ θ θ θ kl kl, gh = kl, = gh l k

3 = θ θ θ agbh, (A.0 gh,,,,, = gh where θ / = θ / ba (, =,..., s used. From (A.9 and (A.0, acov ( θα, = acov ( θα, follows. T T T. Sulement to the aer Asymtotc exansons n mean and covarance structure analyss In ths secton, sulement to Ogasawara (009a s gven, where the subscrts for moments e.g., s, Xa, mcd, μa, cd and κ cd can take values,,, wth ossble dulcaton. A. The asymtotc covarance matrx of samle means and covarances Let s = ( X X ( X X, a a b b = E( X = μ ( =,...,, a esecally a S = ( X μ and a a a = X a = Xa, = d = a μa b μb d μd = S ( X ( X ( X d = E{( Xa μa( Xb μb ( Xd μd } s = S S S a b results:. Then, Cov( Xa, Xb = = O(, S SaSb Sc Cov( s, Xc = E. For comleteness we start wth elementary

4 E( SSc E( SaSbSc c c = = c c c = = O (. It s known that Cov( s, scd = κcd acbd adbc = cd cd ac bd ad bc (see e.g., Kalan, 95, Equaton (, whch gves κ = From the ove results, (.7 follows. cd cd cd O ( A. The thrd cumulants of some samle moments A.. E{( s ( s ( X μ } cd cd e e = E{ s s ( X μ } E{( s ( X μ } cd e e cd cd e e E{( s ( X μ } cd e e E( SScd Se E{( SaSbScd ScSd S Se} = ( ( E( SSSSS ( a b c d e cde cd e = ( cde ( ( cde cd e cde ( { cde cde ( cde cdabe } (. 4

5 5C= 0 cde ( cde cde cde ( cde = ( cde cde ( ( cde cdabe ( O ( cde ( cde cdabe = O (. A.. E{( s ( Xc μc( Xd μd } cd =E{ s( Xc μc( Xd μd } E( SScSd E( SaSbScSd cd = ( ( ( = ( cd cd cd ac bd ad bc cd κ = = cd cd ac bd ad bc cd A.. E{( Xa μa( Xb μb( Xc μc} E( SSS a b c c = =. From A.., A.. and A.., we have (.0a through (.0c. A. The fourth central moments of some samle moments A.. E{( s ( s ( s ( X μ } cd cd ef ef g g. 5

6 =E{ s s s ( X μ } E{ s s ( X μ } cd ef g g cd ef g g E{ s ( X μ }. cd ef g g ( The frst term on the rght-hand sde of (A E( SSS a b cdsefsg E( SSSSS a b c d efsg E( SScd Sef Sg = ( ( ( E( SSSSSSS a b c d e f g 4 ( = ( ( cdefg cdefg cdefg ( ( ( g cd ef (A ( ( ( 4 efg cd cdg ef cdefg ag bcdef bg acdef g cdef acd befg cdg ef efg cd 4 ( ( ( cdgef efgcd acdbgef gcdef 4 6 ( ( ef agbcd cdg ( 4 4 efg cd efa ( bc dg bd cg cd bg O ( cdefg cdefg = ( ( efg cd cdg ef cdefg bcdef ag 6

7 cdefg 4 ( ( acdef bg cdef g befg acd ( ef cdg cd efg g cd ef 4 ( cdgef efgcd acdbgef gcd ef ( 4 6 bcd ag cdg ef efg cd 4 4 efa ( bc dg bd cg cd bg O (. ( The second term on the rght-hand sde of (A E( Scd Sef Sg E{( SSS c d ef SS e f Scd Sg} E( SSSS c d e f Sg = ( ( ( cdefg ( ( cdgef efgcd = ( ( { ( ( cdefg cdgef efgcd } efg cd efc dg efd cg cdg ef cde fg cdf eg ( cde fg efgcd O ( cdefg cdgef efgcd cdefg = ( ( cdgef efgcd efcdg efdcg cde fg cdfeg ( 7

8 (. ( The thrd term on the rght-hand sde of (A = cdefg cde fg efgcd O (v The sum of the terms on the rght-hand sde of (A ( = ( cdefg cdefg bcdef ag acdef bg cdef g 4 cdef g befg acd ( ( * efg ( acbd adbc 4 gcdef ( 4 ( ( 4 4 ( (, efa bcdg bdcg O where the factors ( * ( and ( ** are ** acd bg ef * ( = { ( ( 4( 4 ( } ( O = O ( = O (, ( ( ** ( 4 4 ( = O ( = O (, ( whch gves (A 8

9 ( (A = ( cdefg cdefg bcdef ag acdef bg cdef g 4 4 ( cdef g befg acd g cd ef efg ( acbd adbc 4 acdbgef 4 4 ( ( efa bcdg bdcg O = ( cdef g g cd ef cdefg ( cdefg bcdefag acdefbg 4 cdefg befgacd 4gcdef 4 ( efg ac bd ad bc acd bg ef (A 4 4 efa ( bcdg bdcg O (. From (A, (.5a follows. A.. E{( s ( s ( X μ ( X μ } cd cd e e f f =E{ s s ( X μ ( X μ } E{ s ( X μ ( X μ } cd e e f f cd e e f f E{( X μ ( X μ }. cd e e f f (A4 9

10 ( The frst term on the rght-hand sde of (A4 E( SSS a b cdss e f E( SScdSS e f E( SSSSSS a b c d e f = 4 ( ( ( = { ( ( cdef cdef efcd cdef ( ecdf fcde ( ( cdef } 4 {( ( ( ( ( ( cd ef ef cd cdef acde bf acd bef bcd aef cde f cdf e ef 5 } ( ( ( 4 O ( cdef = ( cdef efcd cdef ( ae bf af be cd 4 cd ef 8 4 acdebf acdbef ecdf fcde ( ecdf fcde ( ( ( cdef aebf afbe cd ( ( ( 5 cdef 4 O (. ( The second term on the rght-hand sde of (A4 E( ScdSeSf E( ScSdSeSf ( ( 0

11 cdef ( cdef = ( { ( ( } cdef cdef cedf cfde ( = ( ( 4 ( (. cedf cfde O ( The thrd term on the rght-hand sde of (A4 cdef = ef cd cdef cd ef (v The sum of the terms on the rght-hand sde of (A4 The coeffcent of cd ef excet that of 5 (/ cdef s = ( ( 4 4 = O ( = O (. ( ( Usng the ove result, (A4 becomes cdef efcd cdef = cdef ( 8 4 ( acde bf e cdf f cde acd bef

12 ( ( e cdf f cde cd ef 5 ( cedf cfde cdef O = ( cdef ecdf fcde cdef ( ( cdef cdef efcd cdef 8 ( acde bf e cdf f cde 4 4 (, 4 ( ( acd bef cd ef ce df cf de 5 cdef O whch yelds (.5b. A.. E{( s ( Xc μc( Xd μd ( Xe μe} E( SSSS c d e E( SSSSS a b c d e E( SSS c d e = 4 ( ( = ( { cde ( ( cde cde} 0 cde 4 cde O ( 0 = 4 (. c de c de cde 4 O (

13 cde 0 4 = (. cde cde cde O ( From the ove equaton, (.5c follows. A..4 E{( X μ ( X μ ( X μ ( X μ } E( SSSS a a b b c c d d a b c d = = ( 4 4 cd cd cd cd cd = O ( whch gves (.5d. A.4 The artal dervatves of 4 (, F = ( x μ' Σ ( x μ F μ ' Σ = Σ ( x μ ( x μ' Σ Σ ( x μ θ θ θ ( =,..., q, F x = Σ x μ (, F μ ' μ ' Σ = Σ ( x μ ( θ θ θ Σ Σ x μ θ θ θ μ ' μ Σ Σ Σ x μ Σ Σ Σ x μ θ θ θ θ ( ' ( Σ ( x μ' Σ Σ ( x μ θ θ (, =,..., q,

14 F Σ μ = Σ Σ ( x μ Σ ( =,..., q, θ x θ θ F μ ' = Σ ( x μ θ θ θ θ θ θ k k μ ' Σ Σ Σ ( x μ θ θ θ k μ ' μ μ Σ Σ ( θ θ Σ Σ x μ θ θ θ θ k k 6 μ ' Σ Σ μ ' Σ Σ Σ ( x μ Σ μ θ θ Σ Σ θ θ θ θ k Σ Σ Σ x μ Σ Σ Σ Σ x μ θ θ θ ( ' ( k Σ Σ ( x μ' Σ Σ Σ ( x μ θ θ θ k Σ ( x μ ' Σ Σ ( x μ (,, k =,..., q, θ θ θ k k F Σ Σ Σ μ = ( θ Σ Σ Σ x μ θ θ Σ Σ x θ θ θ Σ μ Σ Σ ( x μ Σ (, =,..., q, θ θ θ θ F Σ = q k = = θ X X Σ Σ θ (,... ;,,...,, k k 4

15 4 4 F μ' μ = Σ θ θ θ θ θ θ θ θ k l k l μ' Σ μ Σ Σ θ θ θ θ k l 6 μ' μ μ' Σ μ Σ Σ Σ θ θ θ θ θ θ θ θ k l k l μ' Σ Σ μ k l = θ θ θ θ Σ Σ Σ (,,,,...,, k l 4 6 F Σ Σ μ = = Σ Σ Σ θ θ θ k x θ θk θ θ θ Σ Σ μ Σ θ θ θ k Σ μ μ Σ Σ Σ (,, k =,..., q, θ θ θ θ θ θ k k F Σ Σ Σ θ 4 = = θ X k X Σ Σ Σ Σ Σ θ θ θ l θ θ θ kl (, =,... q; k, l =,...,. q A.5 The covarances of some samle moments A.5. Cov( m, X c e E( SS a bcse E( ScSe E( SaSbScSe =E{ mc( Xe μe} = 4 ce aebc κ = O = The ove result yelds (4.7b. A.5. Cov( m, s c ef ce ( O (. 5

16 Frst, we have S S S S E( m E ( ( = c and c a b c c = S asbc = c c c E( SS a bcsef E( ScSef E( SaSbScSef E( ScSeSf E( mcsef = ( ( ( ( ( E( SS a bcss e f E( SSSSS a b c e f 4 ( ( ( aefbc cef cef ( c ef = ( ( cef O ( ( aefbc cef 4cef = cef O From the ove results, (. Cov( mc, sef = cef c ef aef bc O(, whch gves (4.7c. A.5. Cov( m, X cd e 6

17 =E{ m ( X μ } cd e e 4 6 E( SS a bcdse E( SSS a b cdse E( ScdSe E( SaSbScSdSe = ( bcdae cde bcdae cde O O = ( = (. From the ove result, (4.7d follows.. Sulement to the aer: On the estmators of model-based and maxmal rellty In ths secton sulement to Ogasawara (009b s gven. S. The artal dervatves of ρ (=,, 4 and ρ ρ ( δtr( Ψ tr( Ψ =, ( S ' S ' ρ = δ δcd cd ( S ' ( ( tr( Ψ max5 δ tr( Ψ tr( Ψ, (, cd ( S ' cd S ' cd ρ 6 = δ 4 δcd δef Ψ cd ef ( S ' ( ( ( tr( tr( Ψ δ tr( Ψ ( δ ( δcd (, cd, ef ( S ' ef ( S ' cd ef tr( Ψ, S ' cd ef 7

18 ρ ( δ ' ΛΛ ' Λ ' = ' Λ, ( S ' S ' ρ = ( δ ( ' ' δcd ΛΛ cd ( S ' ( Λ' Λ Λ' Λ' (, cd ( S ' cd S ' cd cd δ ' Λ ' Λ, ρ 6 = ( δ ( ( ' ' 4 δcd δef ΛΛ ( S ' cd ef 4 Λ ' ( δ ( ' δcd Λ (, cd, ef ( S ' ef δ Λ Λ ' Λ Λ Λ ' ( S ' scd sef scd s ef S ' s scd sef ( ' ' ' Λ ' ' Λ Λ, S ' cd ef ρ4 tr( Ψ Λ' Ψ tr( Ψ = ', s ( ' Λ ' s s Σ Σ ρ 4 tr( Ψ Λ' Ψ Λ' Ψ = ' ' s scd ( ' Λ s s Λ scd s Σ cd tr( Ψ ' Λ' Ψ Λ (, cd ( ' Σ scd s cd tr( Ψ Λ Λ' Λ' Ψ ' ( ' Λ s scd s scd s s Σ cd tr( Ψ, ' Σ cd 8

19 ρ 4 6tr( Ψ Λ' Ψ Λ' Ψ = ' ' 4 s scd sef ( ' Λ s s Λ scd s Σ cd Λ ' Ψ ' Λ ef ef tr( Ψ Λ' Ψ Λ' Ψ ' ' (, cd, ef ( ' s Λ scd s Λ cd sef s Σ ef tr( Ψ ' ' ' ' Λ Ψ Λ Λ ' Λ Ψ ( ' Λ Λ Σ cd ef cdef cd ef Ψ Λ ' Ψ ' Λ ( ' Σ cd sef s ef tr( Ψ Λ Λ' Λ' Ψ ' Λ ( ' Σ scd sef scd sef scd s ef tr( Ψ ' Λ Λ Λ' Ψ ' Λ ( ' Σ (, cd, ef cdef c d sef s scd s ef tr( Ψ ' Σ cdef ( a b ; c d ; e f. ρ, let y 5 = s / ψ =, then max5 y 5 For max 5 The artal dervatves of ŷ 5 wth resect to y δ s ψ s ρ = {/( }. s are as follows: 5 =, ψa = ψ y 5 δ ψ a s ψ ψ s ψ = (, cd cd ψa cd = ψ cd ψ cd, 9

20 y δ ψ ψ δ ψ 5 a a a = (, cd, ef cd ef ψa cd ef ψa cd ef ( a b ; c d ; e f. 6s ψ ψ ψ s ψ ψ s ψ 4 = ψ (, cd, ef cd ef ψ cdef ψ cdef S. The artal dervatves of Ĥ wth resect to s s / / ( H = Ψ S Ψ for ρ max h h ψ = ψ ψ δ δ ψ ψ (, / / / / s a b, ψ 5/ / ψ / / ψ ψ = ψ ψ ψ ψ cd (, 4 cd 4 cd / / ψ ψ ψ s cd / / ψ ψ / / δδ c d ψ ψ ψ ψ, (, cd h 5 7/ / ψ ψ ψ = ψ ψ (, 8 cd ef cd ef ψ 5/ / ψ 5/ / ψ ψ ψ ψ ψ ψ ψ (, cd, ef 8 cd ef 4 cdef ψ ψ 4 / / ψ ψ cd / / ψ ψ ψ s ef cd ef 40

21 ψ ψ ψ ψ 5/ / / / δδ e f ψ ψ ψ ψ (, cd, ef (, 4 cd 4 cd / / ψ ψ ψ cd ( ; a b ; c d ; e f. ( H = Λ ' S Λ for ρ max h Λ ' δ ( ( = a b, s S Λ (, s S Λ S Λ h ' Λ Λ' Λ = s s S Λ S cd (, cd cd δ Λ ( ( Λ b (, cd S S Λ S Λ a S cd a cd b 8 ac δ δcd s S Λ d S Λ b 4 ( cd,,, ( ( ( (, h ' Λ Λ' Λ = S Λ S cd ef (, cd ef (, cd, ef cd ef δ Λ Λ Λ Λ (, cd, ef S s S S cd a ef S b ef a cd Λ Λ S ( S Λ b ( S Λ a S cd ef a cd ef b b 4

22 6 ac Λ ( δ( δcd s S ( S Λ (,, 4 b cd ef ( a, b, c, d,(, s ef d 48 ae fc ( δ ( ( ( ( δcd δef s s S Λ d S Λ b 8 ( cde,,,,, f ( k ; a b ; c d ; e f, where s = S. ( H = ΛΨ ' Λ for max 4 ( h Λ' Ψ ' =, s Ψ Λ (, Λ Ψ Λ ρ h Λ' Λ' Ψ Λ' Λ = s s Ψ Λ Ψ Λ Ψ cd (, cd (, cd cd cd Ψ Ψ ΛΨ ' Λ Λ ' Ψ Ψ Λ, cd cd h Λ' Λ' Ψ = Ψ Λ s scd sef (, s scd s Ψ Λ ef (, cd, ef cd ef Λ ' Λ Λ' Ψ Λ Λ' Ψ Ψ Ψ Ψ Ψ Λ cd ef cd ef cd ef Λ ' Ψ s s Ψ Ψ Ψ Λ ' s Λ Ψ Λ (, cd, ef cd ef cd ef 4 Ψ Ψ Ψ 6 ' ' Ψ ΛΨ Λ ΛΨ Λ cd ef cd ef ( k ; a b ; c d ; e f. 4

23 S. The thrd artal dervatves of the egenvectors u * k = ( δ h hcd hef = ( γ γ ( γ ( ( γ γ γ γ γ γ γ hcd hef hcd hef ( u u u u u a b b a k ( γ γ ( cd, ef ( γ γ hcd u a u b u k u ( b ua u k uau b ubu a (, hef h ef hef u a ua u b b u γ (, hcd hef hcd he f ( γ u a u b u b u a u k h ef hcd hcd hef u a u b u k u k u ( b ua uau b ubu a ( cd, ef hcd h cd hef hcd hef * * * * ( k, =,... ; a b ; c d ; e f. References Kalan, E. L. (95. Tensor notaton and the samlng cumulants of k-statstcs. Bometrka, 9, 9-. Ogasawara, H. (008. Some roertes of the votal statstc based on the asymtotcally dstrbuton-free theory n structural equaton modelng. Communcatons n Statstcs Smulaton and Comutaton, 7 (0, Ogasawara, H. (009a. Asymtotc exansons n mean and covarance 4

24 structure analyss. Journal of Multvarate Analyss, 00 (5, Ogasawara, H. (009b. On the estmators of model-based and maxmal rellty. Journal of Multvarate Analyss, 00 (6,