Exact inference for the parameter of the exponential distribution based on type-I censored stage life testing data

Journal of Statistical Computation and Simulation, vol. 96, pp. 1459–1496

Abstract

Assuming exponentially distributed lifetimes, we conduct parametric inference for the lifetime distributions when Type-I censoring is implemented on a stage life test experiment as introduced by [Laumen B, Cramer E. Stage life testing. Naval Res Logist (NRL). 2019b;66(8):632.647]. We assume that the lifetime distribution of experimental units follows the exponential distribution in both stages. The distributions under two stress levels are linked by the cumulative exposure model. We derive the maximum likelihood estimators (MLE) for the unknown model parameters and then obtain the conditional moment generating function and the exact distribution of the MLEs. In this paper, we have established the exact conditional distributions of the MLEs by an alternative approach proposed in [Górny J. A New Approach to Hybrid Censoring. PhD thesis, RWTH Aachen University, Aachen, Germany, 2017]. Additionally, we derive exact confidence intervals, asymptotic confidence intervals based on the asymptotic normality of MLEs, and we also employ bootstrap methods to obtain Boot-p and Boot-t confidence intervals. We propose simple approximate confidence intervals based on a χ2- and gamma distribution based approximation, respectively, which are shown to provide very good alternatives even for small sample sizes. Furthermore, we obtain the Bayes estimators (BEs) under the squared error loss function using informative prior for the unknown model parameters. A comparisons of MLEs and BEs are made in terms of their mean squared errors. An extensive simulation study and a data analysis are performed to demonstrate the effectiveness of our approach.

Authors 2

1. Erhard Cramer Aachen Institute of Statistics

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen

   Institute of Statistics, RWTH Aachen, Aachen, Germany

2. Farha Sultana corresponding

   Indian Institute of Information Technology Guwahati

   Affiliation as printed

   Indian Institute of Information Technology Guwahati

   Department of Science and Mathematics, Indian Institute of Information Technology Guwahati, Guwahati, India