Abstract
In this article, estimation of stress-strength reliability $\delta=P\left(Y<X\right)$ based on progressively first failure censored data from two independent inverse Weibull distributions with different shape and scale parameters is studied. Maximum likelihood estimator and asymptotic confidence interval of $\delta$ are obtained. Bayes estimator of $\delta$ under generalized entropy loss function using non-informative and gamma informative priors is derived. Also, highest posterior density credible interval of $\delta$ is constructed. Markov Chain Monte Carlo (MCMC) technique is used for Bayes computation. The performance of various estimation methods are compared by a Monte Carlo simulation study. Finally, a pair of real life data is analyzed to illustrate the proposed methods of estimation.
Highlights
IntroductionThe cumulative distribution function (cdf) of the inverse Weibull distribution (IWD) is given by
The cumulative distribution function of the inverse Weibull distribution (IWD) is given byF (x; α, λ) = e−λx−α, x > 0, α, λ > 0, (1)where α and λ are shape and scale parameters, respectively.The corresponding probability density function and failure rate function, respectively, are given by f (x; α, λ) = αλx−α−1e−λx−α, x > 0, α, λ > 0, (2)αλx−α−1 h (x; α, λ) = eλx−α − 1, x > 0, α, λ > 0. (3)The plot of failure rate function of IWD for different values of shape parameter α and fixed scale parameter λ = 1 is shown in Figure 1. 2.0 α =3.0 λ=1 1.0 h(x)
The generalized entropy loss function (GELF) is proposed by Calabria and Pulcini (1996)
Summary
The cumulative distribution function (cdf) of the inverse Weibull distribution (IWD) is given by. Soliman, Abd-Ellah, Abou-Elheggag, and Modhesh (2012) discussed estimation of the parameters of life for Gompertz distribution using progressive first failure censored data. Some authors have investigated the estimation of δ for some lifetime distributions based on progressively first failure censored data, see for example, Lio and Tsai (2012) They discussed the estimation of stress-strength reliability for Burr XII distribution based on progressively first failure censored sample. Kumar, Krishna, and Garg (2015) studied estimation of stress-strength reliability for Lindley distribution based on progressively first failure censored sample. The main objective of this article is the classical and Bayesian estimation of stress-strength reliability δ = P (Y < X), where, X and Y both are independent random variables having inverse Weibull distributions and samples obtained from both the distribution are progressively first failure censored samples.
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