Semiparametric zero-inflated regression models
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Zero-inflated (ZI) regression models have had wide application recently and have proven useful in modeling data with many zeros. Zero-inflated mixed models (ZIMMs) have been developed to model zero-inflated data with heterogeneity, correlation, or other features not adequately captured by regular ZI models. Semiparametric regression methods have also been implemented in the context of ZI regression recently and can be applied in modeling both the mixing probability and the data from the non-zero state with a combination of parametric and nonparametric components. Although a variety of classes of ZI models have been researched and studied such as the zero-inflated binomial models (Hall, 2000) and zero-inflated negative binomial models, in this dissertation we focus on ZI Poisson (ZIP) models and their mixed-effect extensions, ZIP-mixed models (ZIPMMs), with particular attention on cases involving high-dimensional random effects. In Chapter 2 of this dissertation, we present one approach of fitting ZIPMMs using maximum likelihood (ML) implemented via the Monte Carlo EM (MCEM) algorithm using spherical radial quadrature (Zippunikov and Booth, 2006). The MCEM algorithm is employed to facilitate the computation, and the spherical radial quadrature is applied to evaluate the integral in the Monte Carlo E step. The MLEs of the parameters are obtained at the convergence of the algorithm. In Chapter 3, we propose another approach to fit ZIPMMs using a modified EM algorithm which involves Laplace's approximation (Steele, 1996) to evaluate the integral in the E step. At the convergence of the algorithm, we obtain approximate MLEs. Small-scale simulation studies of the proposed methods are included in Chapters 2 and 3. In addition, in Chapter 4, a more extensive simulation study is conducted in which the properties of these methods are investigated under a wider variety of scenarios and the methods are compared with each other and with alternative approaches for semiparametric ZIP regression from the literature including adaptive Gaussian quadrature (AGQ), thin plate regression splines (Chiogna and Gaetan, 2007), and sieve maximum likelihood (Lam et al., 2006). Chapter 5 presents a discussion and some potential topics for future research.