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Bayesian and Maximum Likelihood Estimation of Stress–Strength Reliability for Mixed Mgamma Records with Application |
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PP: 1427-1441 |
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doi:10.18576/amis/200603
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Author(s) |
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Rana H. Khashab,
Said G. Nassr,
Mohammed Elgarhy,
Mohammed Elgarhy,
Ghareeb A. Marei,
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Abstract |
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| This paper addresses the estimation of the stress–strength reliability parameter R = P(Y < X) when the stress Y and the strength X are independent random variables following the one-parameter Mgamma distribution, a mixture of two gamma components recently proposed for skewed lifetime data. A closed-form representation of R in terms of the regularized incomplete beta function is obtained, and it is shown that, because the beta indices are integers, R is in fact a rational function of the two parameters whose first and second derivatives are available explicitly. Inference is developed under a mixed record scheme in which upper records are observed on the stress variable and lower records on the strength variable. Maximum likelihood estimators are obtained, and Bayesian estimates under non-informative priors are computed with the Tierney–Kadane approximation. All derivatives required by this approximation are given explicitly, including the non-zero mixed derivative of the log-posterior augmented by logR. We also show that the Fisher information about the strength parameter carried by lower records converges to a finite limit as the number of records grows, which governs the finite-sample behaviour of the estimators and of Wald-type intervals; profile-likelihood intervals are proposed and attain close to nominal coverage. Monte Carlo simulations covering three reliability configurations (R ≈ 0.22, 0.50 and 0.76) show that the Tierney–Kadane estimator has smaller mean squared error than the maximum likelihood estimator in the two configurations with R ≥ 0.5, whereas the maximum likelihood estimator is preferable in the low-reliability configuration. An illustrative application to lung cancer survival times demonstrates the methodology. |
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