Some new statistical methods for a class of zero-truncated discrete distributions with applications

Abstract

Counting data without zero category often occurs in various fields. A class of zero-truncated discrete distributions such as the zero-truncated Poisson, zero-truncated binomial and zero-truncated negative-binomial distributions are proposed in literature to model such count data. In this paper, three main contributions have been made for better studying the zero-truncated discrete distributions: First, a novel unified expectation–maximization (EM) algorithm is developed for calculating the maximum likelihood estimates (MLEs) of parameters in general zero-truncated discrete distributions and an important feature of the proposed EM algorithm is that the latent variables and the observed variables are independent, which is unusual in general EM-type algorithms; Second, for those who do not understand the principle of latent variables, a unified minorization–maximization algorithm, as an alternative to the EM algorithm, for obtaining the MLEs of parameters in a class of zero-truncated discrete distributions is discussed; Third, a unified method is proposed to derive the distribution of the sum of i.i.d.zero-truncated discrete random variables, which has important applications in the construction of the shortest Clopper–Pearson confidence intervals of parameters of interest and in the calculation of the exact p value of a two-sided test for small sample sizes in one sample problem.