Abstract
This paper deals with the estimation of reliability R = P[Y < X] when X and Y are two independent random variables with atwo-parameter bathtub shaped failure rate distribution with the samesecond shape parameter. Likelihood and Bayesian methods are proposedto make inferences about R. We obtain the likelihood interval andasymptotic confidence interval for R, and we consider Bayesianpoint estimates of R under both absolute and squared error loss,using either gamma or uniform priors for the three unknown modelparameters. An equal tail Bayesian credible interval for R isinvestigated. Analysis of a real data set is presented forillustrative purposes, and Monte Carlo simulations are performed tocompare: (1) the performance of Bayes estimates under two differentloss functions; and (2) the maximum likelihood and Bayesian methods.
Highlights
Chen (2000) reinvestigates a two-parameter bathtub shaped failure rate distribution which was originally considered by Gurvich et al (1997)
This paper deals with the estimation of reliability R = P[Y < X] when X and Y are two independent random variables with a two-parameter bathtub shaped failure rate distribution with the same second shape parameter
Analysis of a real data set is presented for illustrative purposes, and Monte Carlo simulations are performed to compare: (1) the performance of Bayes estimates under two different loss functions; and (2) the maximum likelihood and Bayesian methods
Summary
Chen (2000) reinvestigates a two-parameter bathtub shaped failure rate distribution which was originally considered by Gurvich et al (1997). The TPBT distribution is the only known two parameter distribution with a bathtub shaped failure rate function that has exact joint confidence regions for the parameters Chen (2000). Sarhan et al (2012) discussed Bayes estimation of the two parameters of the TPBT distribution. Rezae et al (2010) studied the estimation problem for P(Y < X) when X and Y are independent and follow generalized Pareto distributions with common scale parameter. Ali (2013) discussed Bayes estimation of P(Y < X), using different loss functions, when X and Y are independent and Lindley distributed. Kumar et al (2014) studied maximum likelihood maximum likelihood and Bayes estimates for the parameter P(Y < X) when X and Y are independent and Lindley distributed.
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