Estimation of parameters and reliability characteristics for a generalized Rayleigh distribution under progressive type-II censored sample

Abstract

In this article, we obtain maximum likelihood and Bayes estimates of the parameters, reliability and hazard functions for generalized Rayleigh distribution when progressive type-II censored sample is available. Bayes estimates are derived under three loss functions: squared error, LINEX and generalized entropy. It is assumed that the parameters have independent gamma prior distributions. The estimates cannot be obtained in closed form, and hence the method of Lindley’s approximation is employed in obtaining the desired Bayes estimates. The highest posterior density credible intervals of the model parameters are computed using importance sampling procedure. Moreover, approximate confidence intervals are constructed based on the normal approximation to maximum likelihood estimate and log-transformed maximum likelihood estimate. In order to construct the asymptotic confidence interval of the reliability and hazard functions, it is required to find their variances. These are approximated by delta method. A numerical study is performed to compare the proposed estimates with respect to their average values and mean squared error using Monte Carlo simulations. Further, based on the asymptotic normality of the maximum likelihood estimates, we provide the coverage probabilities for some defined pivotal quantities for model parameters. Finally, a real life dataset is considered to compute the proposed estimates.

Keywords:

• Bayes estimates
• Lindley’s approximation
• Progressive type-II censoring
• Confidence interval
• Delta method
• Coverage probability
• Mean squared error
• Importance sampling method

2010 Mathematics Subject Classification:

• 62F10
• 62F15
• 62F40
• 62N01

Acknowledgments

The authors would like to thank the Editor, Associate Editor and two anonymous reviewers for their valuable comments, which have improved the presentation of this article. One of the authors, Kousik Maiti, thanks the financial support provided by the MHRD, Government of India. Suchandan Kayal gratefully acknowledges the partial financial support for this research work under a grant MTR/2018/000350 SERB, India. The authors also wish to thank Dr. Manoj Kumar Rastogi for his valuable suggestions while preparing the draft of this manuscript.