On the existence and uniqueness of the maximum likelihood estimates of the parameters of Birnbaum–Saunders distribution based on Type-I, Type-II and hybrid censored samples

Abstract

The Birnbaum–Saunders (BS) distribution is a positively skewed distribution and is a common model for analysing lifetime data. In this paper, we discuss the existence and uniqueness of the maximum likelihood estimates (MLEs) of the parameters of BS distribution based on Type-I, Type-II and hybrid censored samples. The line of proof is based on the monotonicity property of the likelihood function. We then describe the numerical iterative procedure for determining the MLEs of the parameters, and point out briefly some recently developed simple methods of estimation in the case of Type-II censoring. Some graphical illustrations of the approach are given for three real data from the reliability literature. Finally, for illustrative purpose, we also present an example in which the MLEs do not exist.

Keywords:

• Birnbaum–Saunders distribution
• existence
• hybrid censoring
• maximum likelihood estimate
• Type-I censoring
• Type-II censoring
• uniqueness

2000 AMS Subject Classifications:

• Primary: 62H10
• Secondary: 62G30
• 97K70

Acknowledgements

The research of the first author was supported by an Natural Sciences and Engineering Research Council of Canada Discovery Grant. The authors express their sincere thanks to Prof. Fried, the Editor-in-Chief, and two anonymous reviewers for making some useful comments and suggestions on an earlier version of this manuscript, which led to this improved final version.