Abstract
Geometric Extreme Exponential Distribution (GEE) is one of the statistical models that can be useful in fitting and describing lifetime data. In this paper, the problem of estimation of the reliability R = P(Y < X) when X and Y are independent GEE random variables with common scale parameter but different shape parameters has been considered. The probability R = P(Y < X) is also known as stress-strength reliability parameter and demonstrates the case where a component has stress X and is subjected to strength Y. The reliability R = P(Y < X) has applications in engineering, finance and biomedical sciences. We present the maximum likelihood estimator of R and study its asymptotic behavior. We first study the asymptotic distribution of the maximum likelihood estimators of the GEE parameters. We prove that the maximum likelihood estimators and so the reliability R have asymptotic normal distribution. A bootstrap confidence interval for R is also presented. Monte Carlo simulations are performed to assess he performance of the proposed estimation method and validity of the confidence interval. We found that the performance of the maximum likelihood estimator and also the bootstrap confidence interval is satisfactory even for small sample sizes. Analysis of a dataset has been given for illustrative purposes.
Highlights
The study of stress-strength model is very common in reliability context
We prove that the maximum likelihood estimators and so the reliability R have asymptotic normal distribution
We found that the performance of the maximum likelihood estimator and the bootstrap confidence interval is satisfactory even for small sample sizes
Summary
The study of stress-strength model is very common in reliability context. In this model, we are interested on the inference of the reliability R = P [Y < X], where X is the strength of a component which is subject to the stress Y. The setting that X and Y follow independent exponential distribution was studied by [37, 38], [17], [34] and [8]. The case where X and Y follow bivariate beta, bivariate gamma and bivariate exponential models were considered respectively by [27, 28] and Nadarajah and Kotz [29]. We consider the inference on R = P (Y < X), when X and Y are independent geometric extreme exponential (GEE) distribution random variables with common scale parameter but different shape parameters. We shall denote a geometric extreme exponential model with shape parameter γ and scale parameter β by GEE(γ, β) and the corresponding density function is given by: γβ e−βx f (x; γ, β) = (1 − γe−βx)2 ,.
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