Type I multivariate zero-truncated/adjusted Poisson distributions with applications

Although there is substantial literature on the univariate zero-truncated Poisson (ZTP) distributions, few works have been done for developing multivariate ZTP distributions with dimension larger than two. In this paper, we first propose a new Type I multivariate ZTP distribution to model multivariate zero-truncated count data and develop its important distributional properties, while providing efficient statistical inference methods. The key idea for extending the univariate ZTP distribution to a multivariate version is to identify a new stochastic representation of the ZTP random variable, resulting in a novel expectation-maximization algorithm with a feature that the introduced latent variables are independent of the observed variables. Since the ZTP distribution is closely related with the zero-adjusted Poisson (ZAP) distribution, the second objective of this paper is to propose a new family of Type I multivariate ZAP distributions (including the Type I multivariate ZTP distribution, the Type I multivariate zero-inflated Poisson (ZIP) distribution, the Type I multivariate zero-deflated Poisson (ZDP) distribution as its special cases) by accounting for truncation, inflation and deflation at zero. The corresponding distributional properties are explored and the associated statistical methods are provided. Finally, three real data sets are used to illustrate the proposed distributions and corresponding analysis methods.