COMMENTS: EM-BASED LIKELIHOOD INFERENCE FOR SOME LIFETIME DISTRIBUTIONS BASED ON LEFT TRUNCATED AND RIGHT CENSORED DATA AND ASSOCIATED MODEL DISCRIMINATION

摘要

Left truncation and right censoring often occur in the follow-up studies in various fields, such as life-testing, reliability, biology and medical science. Left truncation gives biased sample, while right censoring produces a type of incomplete data. These issues should be resolved, and so developing statistical methodology based on left truncated and right censored data is very important. However, there is relatively little literature dealing with estimation based on left truncated and right censored data, compared with the literature dealing with estimation based on only censored data. Professors Balakrishnan and Mitra are to be congratulated for this excellent and interesting article (Balakrishnan and Mitra, 2014) addressing EM-based inference based on left truncated and right censored data. One of the important contributions of this article is to provide us elegantly derived EM algorithm steps for most commonly used lifetime distributions, involving the lognormal, Weibull and gamma distributions, and their generalization, the generalized gamma distribution, based on the data design by Hong, Meeker and McCalley (2009). It also discusses asymptotic confidence intervals of the parameters based on the asymptotic normality and bootstrap method. In the context of the asymptotic variance-covariance matrices in the asymptotic normality, the observed information matrices and the Fisher information matrices for the four distributions are nicely derived by using the missing information principle. This article further addresses prediction of lifetime in the case of the lognormal distribution, a model discrimination problem based on Akaike's AIC and Schwarz's BIC, and the future work.