The exponential family of zero-inflated gamma distributions: statistical properties and inference

Abstract

Zero-inflated gamma (ZIG) distributions are mixtures of a degenerate distribution at zero and a gamma distribution, and they are known to have various applications, such as in environmental sciences. Several statistical methods are available, and it is mentioned in the literature that ZIG distributions form an exponential family in the mixture parameter and gamma distribution parameters. However, this fact has not been elaborated further regarding resulting properties in statistical inference. Based on a minimal canonical representation of the regular three-parameter exponential family of ZIG distributions, the associated cumulant function and mean value function are obtained. Existence and uniqueness of the maximum likelihood estimator as well as uniformly minimum variance unbiased estimators for single parameters are studied. Moreover, several uniformly most powerful (unbiased) tests are developed, including optimal tests for hypotheses about the scale parameter of a zero-inflated exponential distribution with unknown mixing proportion. Expressions for the Fisher information matrix and for some divergence measures, such as Kullback–Leibler divergence and Rényi divergences, regarding two ZIG distributions, are stated and utilized to construct asymptotic simultaneous confidence regions for the ZIG parameters. The statistical methodology is illustrated by means of two data sets from the literature.