Semiparametric Regression Based on Zero Altered Models
We develop a class of semiparametric zero-altered models (ZAMs) for count data, estimated through the generalized method of moments (GMM). The proposed framework extends the utility of ZAMs by allowing for both under- and over-dispersion while preserving support on the entire set of natural numbers. This offers an alternative to other models that can experience computational challenges (e.g. Conway-Maxwell-Poisson) or whose supports can be truncated as functions of the parameter space (e.g. generalized Poisson). By formulating estimation through moment restrictions rather than full likelihood specification, the GMM approach provides substantial robustness against model misspecification. In particular, inference based on likelihood methods for parametric generalized linear models (GLMs), such as those built on Poisson, negative binomial, or related count distributions, can become unreliable when the assumed distributional form is violated, often leading to biased estimates and invalid inference. In contrast, our methodology requires only the first two moments of the distribution, ensuring consistent estimation even when higher-order distributional features are misspecified. We establish the asymptotic properties of the estimators, including consistency and asymptotic normality, under general regularity conditions. The flexibility of the approach is illustrated through both simulated experiments and real-data case studies, demonstrating superior performance relative to popular parametric GLMs, particularly in settings characterized by distributional misspecification. To facilitate practical adoption, we provide accompanying R functions for easy implementation, enabling practitioners to apply the proposed methods in applied settings without requiring bespoke programming.