Statistical inference of $P(X<Y)$ for the Burr Type XII distribution based on records

Year 2017, Volume: 46 Issue: 4, 713 - 742, 01.08.2017

Fatih Kızılaslan Mustafa Nadar

Abstract

In this paper, the maximum likelihood and Bayesian approaches have been used to obtain the estimates of the stress-strength reliability $R=P(X<Y)$ based on upper record values for the two-parameter Burr Type XII distribution. A necessary and sufficient condition is studied for the existence and uniqueness of the maximum likelihood estimates of the parameters. When the first shape parameter of $X$ and $Y$ is common
and unknown, the maximum likelihood (ML) estimate and asymptotic confidence interval of $R$ are obtained. In this case, the Bayes estimate
of $R$ has been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to lack of explicit forms under the squared error (SE) and linear-exponential (LINEX) loss functions for informative prior. The MCMC method has been also used to construct the highest posterior density (HPD) credible interval. When the first shape parameter of X and Y is common and known, the ML, uniformly minimum variance unbiased (UMVU) and Bayes estimates, Bayesian and HPD credible as well as exact and approximate intervals of $R$ are obtained. The comparison of the derived estimates is carried out by using Monte Carlo simulations. Two real life data sets are analysed for the illustration purposes.

Keywords

Burr Type XII distribution, Stres-strength model, Record values, Bayes estimation

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