Identifiability of zero-inflated Poisson models

Abstract

Zero-inflated Poisson (ZIP) models, which are mixture models, have been popularly used for count data that often contain large numbers of zeros, but their identifiability has not yet been thoroughly explored. In this work, we systematically investigate the identifiability of the ZIP models under a number of different assumptions. More specifically, we show the identifiability of a parametric ZIP model in which the incidence probability p(x) and Poisson mean λ(x) are modeled parametrically as p(x) = exp(β₀ + β₁x)/[1 + exp(β₀ + β₁x)] and λ(x) = exp(α₀+α₁x) for x being a continuous covariate in a closed interval. A semiparametric ZIP regression model is shown to be identifiable in which (i) p(x) = exp(β₀ + β₁x)/[1 + exp(β₀ + β₁x)] and λ(x) = exp[s(x)], (ii) p(x) = exp[r(x)]/{1 + exp[r(x)]} and λ(x) = exp(α₀ + α₁x), or (iii) p(x) = exp[r(x)]/{1 + exp[r(x)]} and λ(x) = exp[s(x)] for r(x) and s(x) being unspecified smooth functions.