Estimation And Influence Analysis For Generalized Semiparametric Models

In the past ten years, more attraction has been given to mixed-effects models in longitudinal analysis. One of mixed-effects models is generalized partially linear mixed models in which there are both parametric part and nonparametric part. A lot of real problems can be described through this kind of models. Generalized partially linear mixed models are the natural extension of the generalized linear mixed models and semi-parametric linear mixed models. In this paper, we focused on the estimation and diagnosis of generalized partially linear mixed models.The main difficulty in the esimation of generalized partially linear mixed models is how to calculate the conditional expectation. An alternative strategy of solving this problem is to treat the random effects as parameters so that the integrations can be avoided. The approach has been used by Stiratelli, Laird & Ware(1984), Schall(1991), Breslow & Clayton(1993), Lin & Zhang(1999), etc. The central idea of this approach is to use conditional modes rather than conditional means in the score equation(Diggle et al (2002)) which is quite suitable for Gaussian distributed data. However, this approximation tends to break down when there are few pois-son observations in the data set. To overcome these difficulties, McCulloch(1997) generated a new algorithm to estimated the parameters in the generalized linear mixed models, which is named MCNR(Monte Carlo Newton Raphson) algorithm. In MCNR, random effects are treated as missing value and EM algorithm is incoporated when dealing with conditional expections. MCNR not only performs as good as the former approach with normally distributed data, but also shows robustness with poisson data. In this paper, based on the MCNR algorithm proposed by McCulloch(1997), we, extended the algorithm to the generalized partial linear mixed models so that it maight estimate both of the parameters and nonparameters simultaneously. In the new algorithm, we approximate the nonparametric function in generalized partially linear mixed models by P-spline and use GCV to choose the smoothing parameter. Meanwhile, we also give the proofs and the asymptotic properties of the estimators. Finally, in order to test the reliability of the method, the proposed algorithm is illustrated by the simulation and one real data set analysis.Influence analysis and local influence analysis are two major approaches in statistical diagnosis. The former approach is to detect influential points through case deletation while the latter is based on a small perturbation of model. These two strategies have been extended to various kinds of models in the past two decades. For instance, Pregibon(1981) studied the diagosis in Logistic Regression, He Fung and Zhu(2002) considered diagnosis in semiparametric models, Zhu et al(2001) assessed local influence in generalized linear models. On the basis of the Q- function given by Zhu et al(2001) which is associated with the conditional expectation of the complete-data log-likelihood, we generate generalized Cook Distance and generalized DFTT for the parametric and nonparametric part respectively. Four different perturbation schemes are discussed according to Zhu(2003). One real example is presented to prove the methodology.