: 459,. (2011),, Zhu (2008). Y = Xθ + ε, (1.1) Y = (y 1,..., y n ) T, ε = (ε 1,..., ε n ) T, θ = (θ 1,..., θ p ) T, X n p, X i X i, E(ε) = 0, Var (ε)

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1 Chinese Journal of Applied Probability and Statistics Vol.29 No.5 Oct (,, ) (,, ). ;, ;. :,,,. : O ,,,,,, Belsley (1980), Christensen (1992), Critchley (2001). Zhu Lee (2003). Zhu Lee (2001) Zhu (2001) Q., Lee Xu (2004), Xu (2006). Zhu (2007).. Owen (1988, 1991, 2001),,. Owen (1988, 1991, 2001), Kolaczyk (1994), Chen Cui (2003), Qin Lawless (1994), Kitamura (2001).,,, Bartlett,, Hall La Scala (1990), Bootstrap.. Zhu (2008) (NSFC ) (NSFJSBK ) (KYY 11052) (JG13022) ,

2 : 459,. (2011),, Zhu (2008). Y = Xθ + ε, (1.1) Y = (y 1,..., y n ) T, ε = (ε 1,..., ε n ) T, θ = (θ 1,..., θ p ) T, X n p, X i X i, E(ε) = 0, Var (ε) = σ 2 I,,, y i c i, y i, (y i, X T i ), (z i, δ i, X T i ), z i = min(y i, c i ), δ i = I{y i c i }, i = 1,..., n, I{A} A, c i, G.., δ i i y i c i, δ i = 1, y i, δ i = 0, c i.,,,,,.,. 2. x 1,..., x n, x i R d (i = 1,, n), F, F θ = (θ 1,..., θ p ) T Θ, T ( ), F. g(x, θ) = (g 1 (x, θ),..., g r (x, θ)) T, r p, E F F. E F {g(x, θ 0 )} = 0, θ 0 Θ, (2.1) (2.1), { n L E (θ) = sup np i p i = 1, p i > 0, } p i g(x i, θ) = 0. (2.2) Qin Lawless (1994) Owen (2001), p i (θ) = n 1 {1 + t n (θ) T g(x i, θ)} 1, (2.2) L E (θ) = n {1 + t n (θ) T g(x i, θ)} 1, t n (θ) R r, n g(x i, θ){1 + t T n g(x i, θ)} 1 = 0, L E (θ) θ., l E (θ) l E (θ) = n l E,i (θ) = n log{1 + t n (θ) T g(x i, θ)}, l E,i (θ) = log{1 + t n (θ) T g(x i, θ)}, θ, l E (θ) θ, l E ( θ) = sup(l E (θ)), θ Θ

3 460 θ θ (MELE). Q n (t, θ) = n 1 l E (t, θ) = n 1 n l i (t, θ) = n 1 n log(1 + t T g(x i, θ)), l i (t, θ) = log(1 + t T g(x i, θ)), θ t = t n ( θ) Q 1,n (t, θ) = t Q n (t, θ) = n 1 n g(x i, θ){1 + t T n g(x i, θ)} 1 = 0, Q 2,n (t, θ) = θ Q n (t, θ) = n 1 n Qin Lawless (1994) θ t = t n ( θ) n( θ θ0 ) L N(0, C θ ), L, C θ C t θ g(x i, θ)t n {1 + t T n g(x i, θ)} 1 = 0. n( t 0) L N(0, C t ), C θ = (S 21 S 1 11 S 12) 1, C t = S 1 11 S 1 11 S 12S S 21S S 22.1 = S 21 S11 1 S 12 S: ( ) ( S11 S 12 EF (g 2 ) E F ( θ g) T S = S(0, θ 0 ) = = S 21 S 22 E F ( θ g) 0 g = g(x, θ 0 ), θ g = g(x, θ)/ θ ( ), a, a 2 = aa T ) (0,θ 0 ), (2.3),,,.,,.,. Koul (1981),., y ig = δ i z i, i = 1,..., n, (3.1) 1 G(z i ) y i y ig = 1 G(y i ), δ i = 1; 0, δ i = 0, (3.2)

4 : 461 G c i, (Xi T, y ig), i = 1,..., n, E(y ig X i ) = Xi T θ, Y G = Xθ + ε, (3.3) Y G = (y 1G,..., y ng ) T, X θ (1.1), ε 0, (1.1). G, G, G. Susarla Van Ryzin (1980) G Ĝ1: Ĝ 1 (t) = n { 1 + N + (z i ) } I{zi t,δ i =0}, 2 + N + (z i ) N + (z i ) = n I{z j > z i }, I{A} A. Ĝ1(t) j=1 G(t)., Koul (1981) G K-M Ĝ2, I{A} A. Ĝ 2 (t) = 1 n ( n i ) I{zi t,δ i =0}, n i 1 G (z i, δ i, Xi T ), G Ĝ1(t) Ĝ2(t), (3.2) G Ĝ1(t) Ĝ2(t). (3.3) E(y ig Xi T θ X i) = 0, E{X i (y ig Xi T θ)} = 0, g(x i, θ) = X i (y ig X T i θ), (3.4) x i = (x i,1, x i,2 ) = (y i, Xi T ),, r = p, r g(x i, θ), L E (θ) = n {1+t n (θ) T X i (y ig Xi T θ)} 1, t n (θ) R r, n X i (y ig Xi T θ){1 + tt n X i (y ig Xi T θ)} 1 = i (y i, x T i ), (y i, x T i ). Zhu (2008), Cook, Cook ECD i (M) = ( θ(i) θ) T M( θ(i) θ), θ(i) i (y i, x T i ) θ, M, M = θ 2l E(θ) θ= θ,, ELD i (M) = 2{l E ( θ) l E ( θ(i))}.

5 462 ECD i (M) ELD i (M), i. Zhu (2008),, θ(i) t(i) θ(i) = θ n 1 S S 21S11 1 g(x i, θ) {1 + o p (1)}, t(i) = t n 1 (S S 1 11 S 12S S 1 11 )g(x i, θ) {1 + o p (1)}. (3.5) S 11, S 12, S 22, S 22.1, g(x i, θ) = X i (y ig X T i 3.3 ω = (ω 1,..., ω n ) T, Ω, R n. l E (θ ω) = n ω i l E,i (θ), ω 0 Ω, θ, l E (θ ω 0 ) = l E (θ), ω = ω 0,. θ, θ(ω) θ, θ = θ(ω 0 )., (Zhu (2008)) θ). LD E (ω) = 2{l E ( θ) l E ( θ(ω))}. Ω ω 0 h ω(a) = ω 0 + ah, ω(0) = ω 0, h, a. π : η(ω) = (ω T, LD E (ω)) T ω = ω 0 h C h (ω 0 ) = h T H LDE (ω 0 )h, H LDE (ω 0 ) = 2 2 LD E ( θ(ω)) ω ω T ω0 = 2 T { 2 θ l E(θ)} 1 ω0, θ, = 2 LD E (θ, ω)/ θ ω p n, (k, i) θk l E,i (θ). C h (ω 0 ) π ω 0 h, ω h. C h (ω 0 )., H LDE (ω 0 ) H LDE (ω 0 ) = n λ k v k vk T,, λ k H LDE (ω 0 ), v k λ k, λ 1 λ p = = λ n = 0, {v k = (v k1,..., v kn ) T : k = 1,..., n}, H LDE (ω 0 )v k = λ k v k.,, λ 1 v 1. Zhu Zhang (2004), C ei = p m=1 k=1 λ m v 2 mi, i = 1,..., n,

6 : 463 C ei (e i i 1, 0 n ), i (x T i, y i). Zhu (2008), C ei = 2ECD i {1 + o p (1)} = 2ELD i {1 + o p (1)} = 2n 1 T i S i + o p (1), C ei = 2 n ECD i {1 + o p (1)} = 2 n ELD i {1 + o p (1)} = 2p + o p (1), (3.6), i = θ l E,i (x i, θ) = S 21 S11 1 g(x i, θ) + o p (1), g(x i, θ) = X i (y ig Xi T ECD i C ei, i x i. 3.4 (2.1), R i = (R i,1,..., R i,r ) T = g(x i, θ), i = 1,..., n. θ). E F (R i ) 0, n., Zhu (2008),, (σ 2 1,..., σ2 r) = diag{e F (g 2 )}, ( σ 2 1,..., σ2 r), ( Ri s = (Ri,1, s..., Ri,r) s T g1 (x i, = θ),..., g r(x i, θ) ) T. (3.7) σ r 2 ( σ 2 1,..., σ2 r), E F [g k (x i, θ)] n 1 E F { θ g k (x i ) T S S 21S 1 11 g(x i)} 1 n tr[e F { 2 θ g k(x i )}S ]; σ 2 k Var F {g k (x i )} 2n 1 E F {g k (x i ) θ g k (x i ) T S S 21S 1 11 g(x i)} σ 2 1 n 1 E F { θ g k (x i ) T S θg k (x i )},, g(x i ) = g(x i, θ 0 ), g k (x i ) = g k (x i, θ 0 ), k = 1,..., r. j, Ri,j s ( Rs i,j > 3), i. X ik X i k,, E F { θ g k (x i ) T S θg k (x i )} = 1 n { θ g k (x i, θ) T S θg k (x i, θ)} S ij S nij ( t, θ) S 22.1 = S 21 S 1 11 S 12., S n11 = t Q 1,n = 1 n S n12 = θ Q 1,n = 1 n S n22 = θ Q 2,n = 1 n g(x i, θ)g(x i, θ) T (1 + t T g(x i, θ)) 2 θ= θ,t= t ; g(x i, θ)t T θ g(x i, θ) θ g(x i, θ)(1 + t T g(x i, θ)) (1 + t T g(x i, θ)) 2 θ= θ,t= t ; T θ g(x i, θ)tt T θ g(x i, θ) (1 + t T g(x i, θ)) 2 θ= θ,t= t,

7 464, g(x i, θ) = X i (y ig Xi T θ), θg(x i, θ) = X i Xi T., : 1.,. 2. θ θ (MELE). 3. (3.5) (3.6) (3.7) y i = x i + e i, i = 1,..., 150, (4.1) x i U[0, 2], e i χ , x i, e i (4.1) y i, c i E(λ), λ = 1/0.012 ( 16%). (Xi T, z i, δ i ) Xi T = (1, x i ), z i = min(y i, c i ), δ i = I{y i c i }. y[50] = y[50] + 30, y[100] = y[100] 30, (3.1) (Xi T, z i, δ i ) (Xi T, y ig), Xi T = (1, x i ), y ig = δ i z i /(1 G(z i )). (Xi T, y ig) y ig = θ 1 + θ 2 x i + e i, θ MELE θ, C ei R i,1, C ei 2 R i,1

8 : C ei, R i,1, , , ,, 50 30, 50,, , 50..,, ,, 0.05, 0.22, 0.45,,. e i χ , σ = 40, 5, 10 2σ,, 15 2σ,, 30, 2σ,, [5] Stanford, , 69,. 2 : 1. T5 ; 2.. T5, T5 1.0, T T5,.,,,. 1. ; δ, 1, 0 ; x 1 T 5 ; x 2. 24, 24,,. ( ) y = θ 1 + x 1 θ 2 + x 2 θ 3 + ε, y, x 1 x 2 T5. (z i, x 1i, x 2i ), z i = min(y i, c i ), y i, c i, G, x 1i, x 2i i T5.,, (2.3) MELE θ = ( , , ) T,, C ei, R i,3, 3 4.

9 466 1 δ x 1 x 2 δ x 1 x

10 : C ei 4 R i,3 3 C ei, , 4 R i,3, , T5, 39 T5,,.., [1] Belsley, D.A., Kuh, E. and Welsh, R.E., Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, John Wiley, New York, [2] Christensen, R., Pearson, L.M. and Johnson,W., Case-deletion diagnostics for mixed models, Technometrics, 34(1992), [3] Critchley, F., Atkinson, R,A., Lu, G.B. and Biazi, E., Influence analysis based on the case sensitivity function, Journal of the Royal Statistical Society: Series B, 63(2001), [4] Zhu, H.T. and Lee, S.Y., Local influence for generalized linear mixed models, Canadian Journal of Statistics, 31(2003), [5] Zhu, H.T. and Lee, S.Y., Local influence for incomplete data models, Journal of the Royal Statistical Society: Series B, 63(2001), [6] Zhu, H.T., Lee, S.Y., Wei, B.C. and Zhou, J.L., Case-deletion measures for models with incomplete data, Biometrika, 88(2001), [7] Lee, S.Y. and Xu, L., Influence analysis of nonlinear mixed-effects models, Computational Statistics & Data Analysis, 45(2004), [8] Xu, L., Lee, S.Y. and Poon, W.Y., Deletion measures for generalized linear mixed effects models, Computational Statistics & Data Analysis, 51(2006),

11 468 [9] Zhu, H.T., Ibrahim, J.G., Lee, S.Y. and Zhang, H.P., Perturbation selection and influence measures in local influence analysis, The Annals of Statistics, 35(2007), [10] Owen, A., Empirical likelihood ratio confidence intervals for a single functional, Biometrika, 75(1988), [11] Owen, A., Empirical likelihood for linear models, The Annals of Statistics, 19(1991), [12] Owen, A., Empirical Likelihood, Chapman and Hall, New York, [13] Kolaczyk, E.D., Empirical likelihood for generalized linear models, Statistica Sinica, 4(1994), [14] Chen, S.X. and Cui, H.J., An extended empirical likelihood for generalized linear models, Statistica Sinica, 13(2003), [15] Qin, J. and Lawless, J., Empirical likelihood and general estimating equations, The Annals of Statistics, 22(1994), [16] Kitamura, Y., Asymptotic optimality of empirical likelihood for testing moment restrictions, Econometrica, 69(2001), [17] Hall, P. and La Scala, B., Methodology and algorithms of empirical likelihood, International Statisticl Review, 58(1990), [18] Zhu, H.T., Ibrahim, J.G., Tang N.S. and Zhang, H.P., Diagnostic measures for empirical likelihood of general estimating equations, Biometrika, 95(2008), [19],,,,, 27(2011), [20] Koul, H., Susarla, V. and Van Ryzin, J., Regression analysis with randomly right-censored data, The Annals of Statistics, 9(1981), [21] Susarla, V. and Van Ryzin, J., Large sample theory for an estimator of the mean survival time from censored samples, The Annals of Statistics, 8(1980), [22] Zhu, H.T. and Zhang, H.P., A diagnostic procedure based on local influence, Biometrika, 91(2004), Diagnostic Measures for Censored Linear Models based on Empirical Likelihood Method Ding Xianwen (School of Mathematical and Physics, Jiangsu University of Technology, Changzhou, ) Xu Liang (Department of Mathematics, Southeast University, Nanjing, ) In this paper, the diagnostic measures for censored linear models are studied based on the empirical likelihood method. First, the diagnostic measures for linear models are studied; Then, the censored linear models are converted to linear models, and the diagnostic measures for converted models are studied; Last, simulation studies and real data analysis are given to illustrate the validity of statistical diagnostic measures. Keywords: Empirical likelihood, model convert, statistical diagnosis, validity. AMS Subject Classification: 62J20.

Vol. 15 No. 1 JOURNAL OF HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY Feb O21 A

Vol. 15 No. 1 JOURNAL OF HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY Feb O21 A 5 200 2 Vol 5 No JOURNAL OF HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY Feb 200 2 2 50080 2 30024 O2 A 007-2683 200 0-0087- 05 A Goodness-of-fit Test Based on Empirical Likelihood and Application ZHOU