Some Further Issues Concerning Likelihood Inference for Left Truncated and Right Censored Lognormal Data

Abstract

The maximum likelihood estimates (MLEs) of the parameters of a two-parameter lognormal distribution with left truncation and right censoring are developed through the Expectation Maximization (EM) algorithm. For comparative purpose, the MLEs are also obtained by the Newton–Raphson method. The asymptotic variance-covariance matrix of the MLEs is obtained by using the missing information principle, under the EM framework. Then, using asymptotic normality of the MLEs, asymptotic confidence intervals for the parameters are constructed. Asymptotic confidence intervals are also obtained using the estimated variance of the MLEs by the observed information matrix, and by using parametric bootstrap technique. Different confidence intervals are then compared in terms of coverage probabilities, through a Monte Carlo simulation study. A prediction problem concerning the future lifetime of a right censored unit is also considered. A numerical example is given to illustrate all the inferential methods developed here.

Keywords:

• Asymptotic variances
• EM algorithm
• Left truncation
• Lifetime data
• Lognormal distribution
• Maximum likelihood estimators
• Missing information principle
• Parametric bootstrap confidence intervals
• Right censoring

Mathematics Subject Classification:

• 62N01
• 62N02

Acknowledgments

We thank the reviewers for their insightful comments on an earlier version of this article, which resulted in this improved version.