Chinese Journal of Applied Probability and Statistics Vol.28 No.3 Jun (,, ) 应用概率统计版权所用,,, EM,,. :,,, P-. : O (count data)

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1 Chinese Journal of Applied Probability and Statistics Vol.28 No.3 Jun (,, ),,, EM,,. :,,, P-. : O (count data), Poisson Poisson,, (zero-inflation).,.,, ;,,.,, Fahrmeir Echavarrri (2006) ; Yip Yau (2005) ; Xie et al. (2009, 2008) Poisson Score ; Ghosh et al. (2006) Bayesian.,,,.,,,,. Davidian Giltinan (1995), Diggle et al. (2002), Lin Carroll (2001), He et al. (2002), (2006). Zhang et al. (10BTJ001) ( ) ,

2 312 (1998), (2006) Lin Carroll (2001),. : Poisson ; P- ; EM ;. 2., Poisson (Zero-Inflated Generalized Poisson Distribution), ZIGP. Y ZIGP(φ, λ, α), φ + (1 φ)p(0; λ, α), y = 0; P{Y = y} = (1 φ)p(y; λ, α), y > 0, p(y; λ, α) Poisson, p(y; λ, α) = 1 ( λ ) y(1 { + αy) y 1 exp y! 1 + αλ α, λ. Y, Y α 0, Poisson. λ(1 + αy) 1 + αλ E[Y λ, α] = λ, Var [Y λ, α] = λ(1 + αλ) 2, Y ZIGP(φ, λ, α), Y E[Y φ, λ, α] = (1 φ)λ, φ = 0, ZIGP GP. }, y = 0, 1, 2,..., Var [Y φ, λ, α] = (1 φ)λ[(1 + αλ) 2 + φλ, (2.1),, Y ij, y ij, i j i j, t ij, x ij, z ij, i = 1, 2,..., n, j = 1, 2,..., n i, (y ij, x ij, z ij, t ij )., Y ij ZIGP(φ ij, λ ij, α),, α, φ ij λ ij ( Wu Zhang (2006)). logit(φ ij ) = zij T γ + g(t ij) + u i, log(λ ij ) = x T ij β + h(t ij) + v i, (2.2)

3 : 313 γ β z ij x ij, g( ) h( ),, u i v i, (u T i, vt i ) N(0, Σ)., x ij z ij,, (2.1) (2.2) Poisson. 3. P-,,,,,.,, (P- ). Green Silverman (1994), (2.1) (2.2) PL(γ, β, g, h) = l(y; φ, λ) 1 2 λ 1 (g (t)) λ 2 (h (t)) 2 dt, (3.1) λ 1, λ 2 > 0, l(y; φ, λ) = n n i { I(y ij = 0) log [φ ij + (1 φ ij )p(0; λ ij, α)]f(ω i ; Σ)dω i i=1 j=1 + I(y ij > 0) log [1 φ ij p(y ij ; λ ij, α)]f(ω i ; Σ)dω i }, (3.2) ω i = (u T i, vt i ), f(ω i; Σ)., Yu et al. (2002), Ruppert et al. (2003), g(t) = τ 0 + τ 1 t + + τ q t q + K τ q+k (t h k ) q +, k=1 h(t) = δ 0 + δ 1 t + + δ l t l + K δ l+k (t h k ) l +, w+ k = [max(0, w)] k, {h k } K k=1, k=1 τ = (τ 0, τ 1,..., τ q+k ) T, δ = (δ 0, δ 1,..., δ l+k ) T, B 1 (t) = {1, t,..., t q, (t h 1 ) q +,..., (t h k) q + }T, B 2 (t) = {1, t,..., t l, (t h 1 ) l +,..., (t h k ) l +} T, g( ) h( ) g(t) = τ T B 1 (t), h(t) = δ T B 2 (t). Yu et al. (2002), 5 10,, ;, 10,, K = 15.

4 314, Yu et al. (2002) ( g(t)) 2 dt (ḧ(t))2 dt τ T Kτ δ T Kδ, K t ;, Wu Zhang (2006), [rank(k T K)/3],, q = l = EM (2.1) (2.2), (3.1) (3.2),,,,, EM, ω = (u, v). D c = (D o, D m ), D m ω,, D o. PL(γ, β, δ, τ, Σ D c ) PL(γ, β, δ, τ, Σ D c ) = l 1 (φ, λ) + n log f(ω i ; Σ) 1 2 Nλ 1τ T Kτ 1 2 Nλ 2δ T Kδ, (4.1) l(φ, λ) = n N = n n i, λ 1, λ 2. i=1 i=1 j=1 (4.1) EM. E-step: M-step: i=1 n i {I(y ij = 0) log[φ ij + (1 φ ij )p(0; λ ij, α)] + I(y ij > 0) log[1 φ ij p(y ij ; λ ij, α)]}, Q(ψ ψ (m) ) = E[PL(γ, β, δ, τ, Σ D c ) D o, ψ (m) ]; ψ (m+1) = arg max ψ [Q(ψ ψ(m) )]. ψ = (γ, β, δ, τ, Σ), ψ (m) m. E,, Metropolis f ω y ( ), MonterCarlo., f(ω), ω, ω, ω, q(ω, ω ) = min {1, f ω y(ω y, γ, β, τ, δ, Σ)f(ω) } f ω y (ω y, γ, β, τ, δ, Σ)f(ω. ) M,, Σ Σ = 1 J J (ω (j) ) T ω (j), j=1

5 : 315 {ω (j) } J j=1 Metropolis. θ = (γ, β, δ, τ), [ PL(θ, Σ Dc ) ] E = 0, θ Newton-Raphson θ. λ 1, λ 2, GCV(λ 1, λ 2 ) = (2N) 1 Y 1 HY 1 2 {1 (2N) 1 tr(h)} 2, Y 1 = (M T, Y T ), Y T {y ij } (i = 1, 2,..., n, j = 1, 2,..., n i ) N, M T Y T, y ij = 0, M ij = 1, 0. H C = ( B1 0 0 B 2 ) H = SC(C T SC + ) 1 C T,, S = ( S1 0 0 I ), = ( λ1 K 0 0 λ 2 K S 1 = diag(exp{z T ijγ + g(t ij ) + u i }/[1 + exp{z T ijγ + g(t ij ) + u i }] 2 ). λ 1 GCV λ 2, λ 2 = λ 2, GCV λ 1 = λ 1, (λ 1, λ 2 ).. : (1) ψ (0), m = 0; (2) GCV GCV λ (m) 1, λ (m) 2 ; (3) EM ψ (m+1) ; (4) m = m + 1, (2),. 5., 5.1 ),,, Poisson : Y ij ZIP(φ ij, λ ij ), logit(φ ij ) = γ 0 + γ 1 x 1ij + γ 2 x 2i + u i + g(t ij ); log(λ ij ) = β 0 + β 1 x 1ij + β 2 x 2i + v i + h(t ij ),

6 316 i = 1, 2,..., n, j = 1, 2,..., n i, x 1ij U[ 1, 1], x 2i { 1, 1}, t ij U[ 2, 2], (u i, v i ) T N(0, Σ), Σ ( ) ( ) σ11 σ Σ = =. σ 21 σ h(t) = sin(πt/2) 1 + 2t 2, g(t) = sin(πt/2). [sign(t) + 1] β 0 = 1.5, β 1 = 1, β 2 = 2, γ 0 = 2, γ 1 = 0.5, γ 2 = 1, n = 50, n i = 20, 1000, 1. λ , λ , 1, 2 (,, 5% 95% ). 1 β 0 β 1 β 2 γ 0 γ 1 γ t t 1 g(t) 2 h(t) 5.2, ( 10BTJ001 ) ,, 47,,., (y), (x 1 ), BMI(x 2, (kg)/ (m 2 )) (t),, 10, 200.

7 : 317 y, y > 0 Poisson, α = ZIGP, y ij ZIGP(φ ij, λ ij, α), i = 1, 2,..., 10 ( ), j = 1, 2,..., 200 ( ). logit(φ ij ) = γ 0 + γ 1 x 1ij + γ 2 x 2ij + u i + g(t ij ); log(λ ij ) = β 0 + β 1 x 1ij + β 2 x 2ij + v i + h(t ij ). 2, 3, 4. 2 β 0 β 1 β 2 γ 0 γ 1 γ age age 3 g( ) 4 h( ) [1] Fahrmeir, L. and Echavarrri, L.O., Structured additive regression for overdispersed and zero-inflated count data, Applied Stochastic Models in Business and Industry, 22(5)(2006), [2] Yip, K.C.H. and Yau, K.K.W., On modeling claim frequency data in general insurance with extra zeros, Insurance: Mathematics and Economics, 36(2)(2005), [3] Xie, F.C., Wei, B.C. and Lin, J.G., Score test for zero-inflated generalized Poisson mixed regression models, Computational Statistics & Data Analysis, 53(9)(2009), [4] Xie, F.C., Wei, B.C. and Lin, J.G., Assessing influence for pharmaceutical data in zero-inflated generalized Poisson mixed models, Statistics in Medicine, 27(8)(2008), [5] Ghosh, S.K., Mukhopadhyay, P. and Lu, J.C., Bayesian analysis of zero-inflated regression models, Journal of Statistical Planning and Inference, 136(4)(2006), [7] Davidian, M. and Giltinan, D.M., Nonlinear Models for Repeated Measurement Data, Chapman and Hall, [7] Diggle, P.J., Liang, K.Y. and Zeger, S.L., Analysis of Longitudinal Data, Cambridge University Press, 2002.

8 318 [8] Lin, X.H. and Carroll, R.J., Semiparametric regression for clustered data, Biometrika, 88(4)(2001), [9] He, X.M., Zhu, Z.Y. and Fung, W.K., Estimation in semiparametrics model for longitudinal data with unspecificed dependence structure, Biometrika, 89(3)(2002), [10],,,, 25(2)(2009), [11] Zhang, D., Lin, X. and Sower, M.F., Semiparametric stochastic mixed model for longitudinal data, Journal of the American Statistical Association, 93(442)(1998), [12] Wu, H. and Zhang, J.T., Nonparametric Regression Methods for Longitudinal Data Analysis, Wiley Interscience, [13] Green, P.J. and Silverman, B.W., Nonparametric Regression and Generalized Linear Models, Chapman and Hall, [14] Yu, Y. and Ruppert, D., Penalized spline estimation for partially linear single-index models, Journal of the American Statistical Association, 97(460)(2002), [15] Ruppert, D., Wand, M.P. and Carroll, R.J., Semiparametric Regression, Cambridge University Press, Penalized Likelihood Estimation of a Class of Zero-Inflated Semiparametric Mixed-Effect Model Shi Hongxing (School of Primary Education, Chuxiong Normal University, Chuxiong, ) In this paper, we consider an extension of semiparametric linear mixed-effect model to a class of longitudinal data or clustered data with zero-inflation and propose a novel semiparametric mixed model. Based on the maximum penalized likelihood estimation and EM algorithm, we give a method for our proposed model which may estimate both of the parameters and nonparameters simultaneously. In the method, we use GCV to choose the smoothing parameter.finally, we study a simulation analysis and one real data set to illustrate the proposed method. Keywords: Zero-inflation, semiparametric model, penalized likelihood, P-spline. AMS Subject Classification: 62G05.

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Chinese Journal of Applied Probability and Statistics Vol.28 No.3 Jun (,, ) 应用概率统计 版权所用,,, EM,,. :,,, P-. : O (count data)

Chinese Journal of Applied Probability and Statistics Vol.28 No.3 Jun (,, ) 应用概率统计版权所用,,, EM,,. :,,, P-. : O (count data)