Multivariate zero-truncated/adjusted Charlier series distributions with applications

Abstract

Although the univariate Charlier series distribution (Biom. J. 30(8):1003-1009, 1988) and bivariate Charlier series distribution (Biom. J. 37(1):105-117, 1995; J. Appl. Stat. 30(1):63-77, 2003) can be easily generalized to the multivariate version via the method of stochastic representation (SR), the multivariate zero-truncated Charlier series (ZTCS) distribution is not available to date. The first aim of this paper is to propose the multivariate ZTCS distribution by developing its important distributional properties, and providing efficient likelihood-based inference methods via a novel data augmentation in the framework of the expectation-maximization (EM) algorithm. Since the joint marginal distribution of any r-dimensional sub-vector of the multivariate ZTCS random vector of dimension m is an r-dimensional zero-deflated Charlier series (ZDCS) distribution (1[less than or equal to]r<m), it is the second objective of the paper to introduce a new family of multivariate zero-adjusted Charlier series (ZACS) distributions (including the multivariate ZDCS distribution as a special member) with a more flexible correlation structure by accounting for both inflation and deflation at zero. The corresponding distributional properties are explored and the associated maximum likelihood estimation method via EM algorithm is provided for analyzing correlated count data. Some simulation studies are performed and two real data sets are used to illustrate the proposed methods.

Mathematics subject classification primary:62E15; Secondary 62F10