: 13 m, (Y, X ), Y n 1, X = (x 1,, x n ) T n p, t 1, t 2,, t n, N = m n, β p. : y j = x T jβ + ε j, = 1, 2,, m; j = 1, 2,, n. (1.1) φ, corr(ε j, ε k )

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1 Chnese Journal of Appled Probablty and Statstcs Vol.25 No.1 Feb ,2 1 1 ( 1,, ; 2,, ),,. [1] AR(1).,, score,.,. :,,,, score. : O ,. Dggle et al [2],.,,.,. Dggle et al [2], Pnhero and Bates [3], Lard and Ware [4],. [5][6][1],,., Dggle et al [2], Wolfnger [7]., score, score. ( ) (BK ) ( )., E-mal: jgln@seu.edu.cn ,

2 : 13 m, (Y, X ), Y n 1, X = (x 1,, x n ) T n p, t 1, t 2,, t n, N = m n, β p. : y j = x T jβ + ε j, = 1, 2,, m; j = 1, 2,, n. (1.1) φ, corr(ε j, ε k ) = φ (j k), ε = (ε 1, ε 2,, ε n ) T, Var (ε ) = σ 2 [I n + φ (J n I n )], I n n n, J n n n 1. [1], σ 2, Cook and Wesberg [8], σ 2 j = σ2 m j = σ 2 m(z j, γ), γ 0 m(z j, γ 0 ) = 1, M = dag(m 1, m 2,, m n ), V = I n + φ (J n I n ), ε N(0, σ 2 M V ). φ, Núñez-Antón and Zmmerman [9], φ = φw(v, γ), γ 0 w(v, γ 0 ) = c 0, V = I n + φw(v, γ)(j n I n ). 2 ; 3,, ( M = I n ),. θ = (φ, β T, σ 2 ) T, θ : l(θ) = C N 2 log σ2 1 2 log V 1 ε T V 1 ε. (2.1) C, φ, ; V = I n + φ(j n I n ), ε = Y X β., : 2.1 H 0 : φ = 0; H 1 : φ 0. (2.2) (1.1), (2.2) score : SC φ = 1 2 σ 4 {ut 1 1 m (1 T mu 2 1 m ) 1 1 T mu 1 } θ. (2.3) U 2 = dag{tr((j n I n ) 2 )}, 1 m = (1, 1,, 1) T, u }{{} 1 = (u 11, u 12,, u 1m ) T, u 1 = m n k,l=1;k l ε k ε l.

3 14 : score, l(θ) : l(θ) φ = 1 2 l(θ) = 1 β k σ 2 tr(v 1 ε T V 1 X k, V ) + 1 l(θ) σ 2 ε T V 1 V V 1 ε, V = V φ, = N + 1 2σ 4 ε T V 1 ε., I Y (θ) = E θ [ 2 l(θ)/ θ θ T ],, U 1 = dag{tr(v 1 (J n I n ))}, U 2 = dag{tr(v 2 (J n I n ) 2 )}, V = dag{v }, X = (X1 T,, XT m) T, θ Fsher : I Y (θ) = 1 2 1T mu 2 1 m T mu 1 1 m 0 1 1T mu 1 1 m 1 σ 2 XT V 1 X 0 l(θ)/ φ, H 0 : φ = 0 score : l( θ)/ φ = [1/(2 σ 2 )] 1 T mu 1., H 0, I 1 Y (θ), φ : Iφφ =2(1 T mu 2 1 m ) 1. Cox and Hnkley [10], (2.2) score : SC φ = [( l(θ) φ N 2σ 4, ) ( T I φφ l(θ) )] φ θ. (2.4) θ θ H 0, l( θ)/ φ, I φφ (2.4) H 0 : φ = 0 score : SC φ = 1 2 σ 4 {ut 1 1 m (1 T mu 2 1 m ) 1 1 T mu 1 } θ. Score (score test statstc).. Cox and Hnkley [10], [11]. Score,,., score,.,. (2.3) (2.2), SC φ,. θ R s (s θ ) Θ, ξ = (φ, σ 2 ) T, θ = (0, β T, σ 2 ) T θ H 0. θ 0 = (0, β T 0, σ 2 0 )T Θ φ = 0 θ. β Vonesh and Carter [12]. : θ 0 θ, m,

4 : 15 () m 1 m X T (I n + φ(j n I n )) 1 X A > 0, m 1 I ββ (θ) J ββ (θ) > 0; () {n }, m 1 I σ 2 σ 2(θ) J σ 2 σ2(θ) > 0; () m 1 I ξξ (θ) J ξξ (θ) > 0; (v) m 1 [I φφ (θ) I φσ 2(θ)I 1 σ 2 σ 2 (θ)i φσ 2(θ) T ] J φφ (θ) J φσ 2(θ)J 1 σ 2 σ 2 (θ)j φσ 2(θ) T > 0,, I ab (θ) I Y (θ) ( ) (a, b).. 2.2, H 0 SC φ χ 2 (1). :,, : φ = 0, (A1) θ P θ 0 ; L N(0, J 1 (A2) m 1/2 ( β β 0 ) m 1/2 l(θ)/ ξ θ0 L N(0, J 1 σ 2 σ 2 (θ 0 )); (A4) ββ (θ 0)); (A3) m 1/2 ( σ 2 σ0 2) L N(0, J 1 ξξ (θ 0)); l(θ)/ φ θ (0, β T, σ0 2) : l(θ) φ = l(θ) θ φ + 2 l(θ) (0, (0, βt, σ 2 0 ) T φ σ 2 βt, σ 2 ) ( σ2 σ0). 2 (2.5) T σ 2 = α σ 2 + (1 α)σ0 2 P, 0 < α < 1, φ = 0, σ2 m 1 2 l(θ)/( φ σ 2 P ) (0, βt, σ 2 Jφσ 2(θ 0 ), 0 )T σ 2 0, (0, β T, σ 2 0 )T P θ0, l(θ) σ 2 θ = l(θ) σ 2 (0, βt, σ 2 0 ) T + 2 l(θ) σ 2 σ 2 (0, βt, σ 2 ) T ( σ2 σ 2 0) = 0. σ 2 = α σ 2 + (1 α )σ 2 0, 0 < α < 1. (0, β T, σ 2 ) T P P θ0, m 1 2 l(θ)/( σ 2 σ 2 ) Jσ (0, βt, σ 2 ) T 2 σ 2(θ 0), m 1/2 ( σ 2 σ0 2 1 ) J (θ σ 2 σ 2 0 )m 1/2 l/ σ 2, (0, βt, σ 2 0 ) T, (2.5) : 1/2 l(θ) m φ J (0, β T 0, σ0 2 φσ 2(θ 0)J 1 1/2 l(θ) (0, (θ )T σ 2 σ 2 0 )m σ 2 β T 0, σ0 2 ( )T = [1, J φσ 2(θ 0 )J 1 1/2 l(θ) (θ σ 2 σ 2 0 )] m ). ξ θ0 (A4) φ = 0, m 1/2 N(0, (J l(θ)/ φ θ φφ ) 1 (θ 0 )), (J φφ ) 1 (θ 0 ) = J φφ (θ 0 ) J φσ 2(θ 0 )J 1 (θ σ 2 σ 2 0 )Jφσ T (θ 2 0 ). mi φφ P J φφ (θ 0 ), θ, m, SC φ = [( m 1/2 l(θ) φ L ) T (mi φφ )( m 1/2 l(θ) φ )] θ L χ 2 (1).

5 ,,. Núñez-Antón and Zmmerman [19], φ φ = φw(v, γ), (3.1), v, φ, = 1, 2,, n, 1 1, γ 0,, w(v, γ 0 ) = c 0, c. : H 0 : γ = γ 0 ; H 1 : γ γ 0. (3.2) θ = (γ T, β T, φ, σ 2 ) T V = I n + φw(v, γ)(j n I n ), γ q,. θ (2.1). 3.1 (1.1), (3.2) score : SC γ1 = 1 2 σ 4 {(u 2 σ 2 U 3 1 m ) T Ẇ [Ẇ T (A A1 m (1 m T A1 m ) 1 1 m T A)Ẇ ] 1 Ẇ T (u 2 σ 2 U 3 1 m )} θ, (3.3) U 3 =dag{tr(v 1 (J n I n ))}, U 4 =dag{tr(v 2 (J n I n ) 2 )}; Ẇ =[ w(v, γ)/ γ k ] m q, u 2 = (u 21, u 22,, u 2m ) T, u 2 = ε T 1 V (J n I n ) V 1 ε, A = U 4 U 3 (1 m 1 T m/n) U 3. : score, 2.1, l(θ),, I Y (θ) = E θ [ 2 l(θ)/ θ θ T ], W = (w(v, γ)) m 1,, θ Fsher : φ 2 2 Ẇ T φ U 4 Ẇ 0 2 Ẇ T φ U 4 W Ẇ T U 3 1 m 1 0 I Y (θ) = σ 2 XT V 1 X 0 0 φ 2 W T 1 U 4 Ẇ 0 2 1T mw T U 4 W 1, W T U 3 1 m φ 1 N 1T mu 3 Ẇ 0 1T mu 3 W 2σ 4 l(θ)/ γ, H 0 : γ = γ 0 score : l( θ) γ = φ 2 Ẇ T U 3 1 m + φ 2 σ 2 Ẇ T u 2 = φ 2 σ 2 [Ẇ T (u 2 σ 2 U 3 1 m )] θ., H 0, I 1 Y (θ), γ : { φ I γγ 2 = 2 Ẇ T [A A1 m (1 T ma1 m ) 1 1 T ma]ẇ } 1.

6 : 17 Cox and Hnkley [10], (3.2) score : SC γ1 = [( l(θ) γ ) ( T I γγ l(θ) )] γ θ. (3.4) l( θ)/ γ, I γγ H 0 : γ = γ 0 score (3.3). H 0 : γ = γ 0, SCγ. θ R s (s θ ) Θ, ξ = (γ T, φ, σ 2 ) T, η = (φ, σ 2 ) T, θ = (γ T 0, β T, φ, σ 2 ) T θ H 0. θ 0 = (γ T 0, βt 0, φ 0, σ 2 0 )T Θ γ = γ 0 θ, : θ 0 θ, m, () m 1 m X T (I n + φw(v, γ)(j n I n )) 1 X A(θ) > 0, m 1 I ββ (θ) J ββ (θ) > 0; () m 1 I ηη (θ) J ηη (θ) > 0; () m 1 I ξξ (θ) I ξξ (θ) > 0; (v) m 1 [I γγ (θ) I γη (θ)iηη(θ)i 1 γη (θ) T ] J γγ (θ) J γη (θ)jηη(θ)j 1 γη (θ) T > 0;, I ab (θ) I Y (θ) ( ) (a, b). 2.2,, H 0 SC γ1 χ 2 (q). H 0, SC γ1 χ 2 (q). 3.2, φ φ, σj 2, (Cook and Wesberg [8] ), : σ 2 j = σ 2 m j = σ 2 m(z j, γ). (3.5) z j, m(, ) γ γ 0, = 1, 2,, m, j = 1, 2,, n, m(z j, γ 0 ) = 1, H 0 : γ = γ 0. θ = (γ T, β T, φ, σ 2 ) T, θ (2.1) : l h (θ) = C N 2 log σ2 1 2 log M V 1 ε T V 1 M 1 ε. (3.6) l h (θ). l h (θ) : l h (θ) = 1 γ k 2 l h (θ) = 1 β k σ 2 tr(m 1 Ṁ k ) + 1 ε T V 1 M 1 X k, ε T V 1 M 1 Ṁ k M 1 ε, Ṁ k = M, γ k

7 18 : l h (θ) φ = 1 2 l h (θ) σ 2 tr(v 1 = N + 1 2σ 4 V φ ) + 1 ε T V 1 M 1 ε. ε T V 1 V φ V 1 M 1 ε, Vφ = V φ,, θ Fsher I Y (θ) I γγ (k,l) = 1 2 I (k) = 1 γσ 2 I φφ = 1 2 I φσ 2 = 1 tr(ṁkṁl), I (k) γφ = 1 2 tr( V φ V 1 Ṁ k ) = I φγ (k), tr(ṁk) = I (k) σ 2 γ, I ββ = 1 σ 2 XT V 1 X, tr( V φ V 1 V φ V 1 ), I σ 2 σ 2 = N 2σ 4, tr( V φ V 1 ) = I σ 2 φ, Ṁk, Ṁ l γ 0. H 0 : γ = γ 0, θ Fsher : I γγ 0 I γφ I γσ 2 0 I ββ 0 0 I Y (θ) =. I φγ 0 I φφ I φσ 2 I σ 2 γ 0 I σ 2 φ I σ 2 σ 2 I γγ = (I γγ I 1 I 1 2 IT 1 ) 1, (3.7) I γγ I Y (θ) I 1 Y (θ) I γγ, ( ) Iφφ I φσ 2 I 1 = [I γφ, I γσ 2], I 2 =. I σ 2 φ I σ 2 σ 2 l h (θ)/ γ k score :. l h ( θ) γ = { 1 2 tr(ṁk) + 1 ε T V 1 Ṁ k ε }q 1. l h ( θ)/ γ, I γγ (3.4) score SC γ2, χ 2 (q) 3.3,, (3.1) (3.5). Zhang and Wess [19],, γ

8 : 19, (3.1) (3.5) γ.,, (3.2). θ = (γ T, β T, φ, σ 2 ) T,, γ, q,. θ (3.6)., H 0, score l h ( θ) γ = { 1 2 tr(ṁk + V 1, H 0, I (k,l) γγ = 1 2 I (k) γφ = 1 2 I φφ = 1 2 tr(v 1 tr(v 1 tr(v 1 V k ) + 1 V k V 1 V k V 1 V k + Ṁk). V l + 2V 1 V φ + V 1 ε T (V 1 V k V 1 V k Ṁ l + ṀkṀl), V φ Ṁ k ) = I (k) φγ, + V 1 Ṁ k )ε }q (3.4) score SC γ3, χ 2 (q). 4. Tsa [13],,,, φ = 0, σ 2 j σ2. Tsa [13] ; [14] Tsa [13] ; [1].. ψ = (γ T, φ) T, ψ 0 = (γ T 0, 0)T, H 0 : ψ = ψ 0. 3, l c (θ). 3 l/ γ k l/ φ, score : l c (θ) ψ = ( 1 2 tr(ṁk) + 1 ) ε T Ṁ k ε 1 tr( 2 V φ ) + 1 ε T V φ ε q 1, I Y (θ), H 0, I ψψ = [I ψψ (θ) I ψσ 2I 1 σ 2 σ 2 I T ψσ 2 ] 1 = ( Iγγ I γσ 2I 1 σ 2 σ 2 I T γσ 2 I φγ I γφ I 1 φφ )

9 20 (I ψψ θ Fsher I(θ) = E[ 2 l(θ)/ θ θ T ] I 1 (θ) ψ ),, I (k,l) γγ = 1 2 I (k) γφ = 1 2 I φφ = 1 2 tr(ṁkṁl), tr( V φ Ṁ k ) = I (k) φγ, tr( V φ Vφ ). Ṁk, Ṁ l, V φ ψ 0,. Cox and Hnkley [10], H 0 : ψ = ψ 0 score : SC ψ = [( lc (θ) ψ ) ( T I ψψ l c (θ) )] ψ θ, (4.1) θ = (ψ T 0, β, σ 2 ) T θ H 0. (4.1) score SC ψ. χ 2 (q + 1)., H 0 SC ψ 5., Zerbe [15], Ch and Resel [16] AR(1), Pan and Fang [17] : 33, 0, 0.5, 1, 1.5, 2, 3, 4, 5.,. 13,, : Y = X T β + ε, ε N(0, σ 2 M V ), = 1, 2,, 13; (5.1) Y = (y j ), j = 1, 2,, 8; y j j, Pan and Fang [17] X = : (a) ; (b) ; (c) ; (d)

10 : 21 ; (e). m j = exp(z j γ) ( z = (z j ) T = (0, 0.5, 1, 1.5, 2, 3, 4, 5) T ) ; w = exp(v γ)/[1 + exp(v γ)], v., γ = 0, m j = 1 w = 1/2,., H 0 : γ = 0; H 1 : γ 0. (a), (d), φ = 0, β, σ 2 1, (b), (c) (e), β, σ 2 φ 2. 1 (a) (d) β 1 β 2 β 3 β 4 σ (b), (c) (e) β 1 β 2 β 3 β 4 σ 2 φ , score, 3. 3 (a) (b) (c) (d) (e) score df p ,,. Vonesh and Chnchll [18],, AIC (Akake s nformaton crteron) SBC (Schwarz s Bayesan nformaton crteron). AIC SBC, 4., (a) (d) AIC SBC,, (a) (d),. 4 AIC SBC (a) (b) (c) (d) (e) df AIC SBC

11 22 6. : () ; () ; () ; (v). : y j = β 1 + x j β 2 + x 2 jβ 3 + x 3 jβ 4 + ε j, = 1, 2,, m, j = 1, 2,, n, m, n, ε N(0, σ 2 M V ), V = I n + φ (J n I n ), M = dag(m 1, m 2,, m n ). φ m j : m j = exp{z j γ}, φ = φ exp{v γ} ; = 1, 2,, m, j = 1, 2,, n. 1 + exp{v γ}, m n, SC, 0.05,, H 0, 1000,,. 1. () : (1) {x j, = 1, 2,, m, j = 1, 2,, n}, [0,5]. φ σ 2 = 0.3, N(0, σ 2 ) {ε j, = 1, 2,, m, j = 1, 2,, n}. (2) β 1 = 4, β 2 = 2, β 3 = 0.7, β 4 = 0.1 {y j, = 1, 2,, m, j = 1, 2,, n}. (3) m n, φ, power (a) φ m=20, n=20 m=30, n=20 m=50, n=30 1

12 : 23, (m, n) (20,20), (30,20), (50,30), 1. α = (0.048,0.051). (m, n) (20, 20),. H 0 : φ = 0, φ,,,.,, (m, n) (20,20) (30,20),, φ < 0 φ > 0, (m, n) (50,30),, φ < 0 φ > () (v) : (1) 1 {x j }, γ σ 2 = 0.3 {ε j }. (2) {z j }, [0,5]. γ, m j = exp{z j γ} {m j }. (3) {v, = 1, 2,, m}, [0,13]. γ, φ ( = 1, 2,, m). (4) β 1 = 4, β 2 = 2, β 3 = 0.7, β 4 = 0.1 {y j }. (5) m n, γ, ; 3., H 0 : γ = 0, 0.05,. γ,. γ. () (v),. power (c) m=10, n=5 m=13, n=8 m=15, n= γ 2

13 24 (e) power m=10, n=8 m=13, n=8 m=20, n= γ 3 3. () : (1) {x j }, φ, γ σ 2 = 0.3 {ε j }, {m j }. (2) β 1 = 4, β 2 = 2, β 3 = 1, β 4 = 0.1 {y j }. (3) m n, φ γ, 1000., (m, n) (13,8), (20,10), H 0 : ψ = ψ 0, 0.05,. H 0 : ψ = ψ 0, ψ φ γ,,, (m, n) γ\φ (13,8) (20,10)

14 : 25 7., score,., score. ( AIC SBC).,,... [1],,,, 27(3)(2004), [2] Dggle, P.J., Heagerty, P., Lang, K.-Y. and Zeger, S.L., Analyss of Longtudnal data, New York, Oxford Unversty Press, [3] Pnhero, J.C. and Bates, D.M., Mxed-Effects Models n S and S-PLUS, New York, Sprnger-Verlag, [4] Lard, N.M. and Ware, J.H., Random-effects models for longtudnal data, Bometrcs, 38(4)(1982), [5],,,, 17(1)(2004), [6],,,, 22(2)(2002), [7] Wolfnger, R.D., Heterogeneous varance-covarance structures for repeated measures, Journal of Agrcultural, Bologcal and Envromental Statstcs, 1(2)(1995), [8] Cook, R.D. and Wesberg, S., Dagnostc for heteroscedastcty n regresson, Bometrka, 70(1983), [9] Núñez-Antón, V. and Zmmerman, D.L., Modellng nonstatonary longtudnal data, Bometrcs, 56(2000), [10] Cox, D.R. and Hnkley, D.V., Theoretcal Statstcs, London, Chapman and Hall, [11],,,,,, [12] Vonesh, E.F. and Carter, R.L., Mxed-effects nonlnear models regresson for unbalanced repeated measures, Bometrcs, 48(1992), [13] Tsa, C.L., Score test for the frst-order autoregressve model wth heteroscedastcty, Bometrka, 73(1986), [14],,,, 4(1994), [15] Zerbe, G.O., Randomzaton analyss of the completely randomzed desgn extended to growth and response curves, Journal of the Amercan Statstcal Assocaton, 74(1979),

15 26 [16] Ch, E.M. and Rensel, G.C., Models for longtudnal data wth random effects and AR(1) errors, Journal of the Amercan Statstcal Assocaton, 84(1989), [17] Pan, J.X. and Fang, K.T., Growth Curve Models and Statstcal Dagnostcs, New York, Sprnger- Verlag, [18] Vonesh, E.F. and Chnchll, V.M., Lnear and Nonlnear Models for the Analyss of Repeated Measurements, New York: Marcel Dekker, Inc., 1997, [19] Zhang, F. and Wess, R.E., Dagnosng explanable heterogenety of varance n random-effects models, Canad. J. Statust., 28(2000), Testng for Homogenety of Varance and Correlaton Coeffcents n Unform Correlaton Models Based on Longtudnal Data Fan Junhua 1,2 Ln Jnguan 1 We Bocheng 1 ( 1 Department of Mathematcs, Southeast Unversty, Nanjng, ) ( 2 School of Chengxan, Southeast Unversty, Nanjng, ) In longtudnal data analyss, homogenety of varance s a basc assumpton. However, ths assumpton s not necessarly approprate. Ln and We [1] consdered the tests for homogenety of wthn-ndvdual varances and between-ndvdual autocorrelaton coeffcents n nonlnear models wth AR(1) errors based on longtudnal data. Ths paper dscusses the tests for homogenety of varances and correlaton coeffcents n longtudnal data model wth unform correlaton covarance structure and obtans several score test statstcs. The glucose data s used to llustrate our results. Power smulatons of the proposed tests are gven n ths paper. Keywords: Longtudnal data, unform correlaton model, heteroscedastcty, correlaton coeffcent, score test. AMS Subject Classfcaton: 62J25.

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! Ν! Ν Ν & ] # Α. 7 Α ) Σ ),, Σ 87 ) Ψ ) +Ε 1)Ε Τ 7 4, <) < Ε : ), > 8 7 !! # & ( ) +,. )/ 0 1, 2 ) 3, 4 5. 6 7 87 + 5 1!! # : ;< = > < < ;?? Α Β Χ Β ;< Α? 6 Δ : Ε6 Χ < Χ Α < Α Α Χ? Φ > Α ;Γ ;Η Α ;?? Φ Ι 6 Ε Β ΕΒ Γ Γ > < ϑ ( = : ;Α < : Χ Κ Χ Γ? Ε Ι Χ Α Ε? Α Χ Α ; Γ ;