Semiparametric regression analysis of zero-inflated data

Zero-inflated data abound in ecological studies as well as in other scientific and quantitative fields. Nonparametric regression with zero-inflated response may be studied via the zero-inflated generalized additive model (ZIGAM). ZIGAM assumes that the conditional distribution of the response variable belongs to the zero-inflated 1-parameter exponential family which is a probabilistic mixture of the zero atom and the 1-parameter exponential family, where the zero atom accounts for an excess of zeroes in the data. We propose the constrained zero-inflated generalized additive model (COZIGAM) for analyzing zero-inflated data, with the further assumption that the probability of non-zero-inflation is some monotone function of the (non-zero-inflated) exponential family distribution mean. When the latter assumption obtains, the new approach provides a unified framework for modeling zero-inflated data, which is more parsimonious and efficient than the unconstrained ZIGAM. We develop an iterative algorithm for model estimation based on the penalized likelihood approach, and derive formulas for constructing confidence intervals of the maximum penalized likelihood estimator. Some asymptotic properties including the consistency of the regression function estimator and the limiting distribution of the parametric estimator are derived. We also propose a Bayesian model selection criterion for choosing between the unconstrained and the constrained ZIGAMs. We consider several useful extensions of the COZIGAM, including imposing additive-component-specific proportional and partial constraints, and incorporating threshold effects to account for regime shift phenomena. The new methods are illustrated with both simulated data and real applications. An R package COZIGAM has been developed for model fitting and model selection with zero-inflated data.