EM and MM algorithms for a class of left-truncated discrete models

Abstract

Left-truncated count data often occur in various fields. Examples include working years of Chartered financial analyst certificate holders, numbers of quinsy of patients who had tonsillectomy, numbers of student enrollments for English course in the University of Windsor, numbers of research papers accepted to an SCI or SSCI journal of a student enrolled in the PhD Program in Applied Statistics in Feng Chia University and so on. A class of left-truncated discrete models such as left-truncated Poisson, left-truncated binomial, left-truncated negative binomial, left-truncated generalized Poisson distributions are proposed in the literature to model such count data. However, the estimates of the parameters in such distributions may be difficult to obtain since the original data set is left-truncated, and hence it is incomplete. For the case of no covariates, in this thesis, I first develop a novel expectation-maximization (EM) algorithm via the stochastic representation method for calculating the maximum likelihood estimates (MLEs) of parameters in the general left-truncated discrete distributions including the general zero-truncated discrete distributions as special cases. 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. Next, I propose a unified minorization-maximization algorithm for obtaining the MLEs of parameters in these left-truncated discrete distributions, since their closed-form solutions are not available in the M-step of the EM algorithm for some distributions. In addition, Bayesian approaches are also developed for the left-truncated Poisson and the left-truncated binomial distributions, respectively.

To incorporate the existence of covariates, I furthermore introduce the left-truncated Poisson regression model and the left-truncated binomial regression model, and utilize De Pierro's algorithm to derive the MLEs of the regression coefficients. The performances of all the proposed methods in this thesis are evaluated through simulation studies and three real data sets are analyzed to illustrate the proposed methods.