Statistics Analysis For Semiparametric Zero-Inflated Negative Binomial Regression Models

Count data observed much larger zeros than expected are often involved in many fields, including public health, epidemiology, sociology, psychology, engineering, agriculture and others. Then Zero-Inflated Poisson (ZIP) regression models are widely used to analysis such data. And some authors presented, in practice, ZI count data are often over-dispersed and ZI-negative binomial (ZINB) distribution may be more appropriate than ZIP distribution. Therefore, the present paper focus on the statistics analysis for semi-parametric ZINB regression models. This kind of model not only contains parametric component but nonparametric component and can be used to describe more practical problems. Influence diagnostics have become part of any serious statistical analysis. This article develops influence diagnostics for semi-parametric ZINB regression models based on the global influence analysis and local influence analysis. Maximum penalized likelihood estimators (MPLEs) for both linear coefficients and nonparametric function are obtained. Then the one-step approximations of the MPLEs in the case-deletion model (CDM) are given and case-deletion measures, such as Generalized Cook’s distance, WK measures and Likelihood distance, are obtained. Meanwhile, it is shown that the CDM is equivalent to the mean shift outlier model (MSOM) in semi-parametric ZINB regression models and outlier tests are presented based on the MSOM. Furthermore, the normal curvatures of local influence are derived under various perturbation schemes including case-weights perturbation and explanatory variable perturbation. At the same time, we discuss score tests for homogeneity of a in semi-parametric ZINB regression models and for testing ZIP regression models against ZINB alternative. Some simulated examples are given to illustrate our methodology and the properties of score test statistics obtained in this paper are investigated through Monte Carlo simulations.