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Estimation of \(R=P(Y<X)\) for three-parameter Weibull distribution. (English) Zbl 1169.62012
Summary: We consider the estimation of the stress-strength parameter \(R=P(Y<X)\), when \(X\) and \(Y\) are independent and both are three-parameter Weibull distributions with common shape and location parameters but different scale parameters. It is observed that the maximum likelihood estimators do not exist in this case, and we propose a modified maximum likelihood estimator, and also an approximate modified maximum likelihood estimator of \(R\). We obtain the asymptotic distribution of the modified maximum likelihood estimators of the unknown parameters which can be used to construct confidence interval of \(R\). Analyses of two data sets have also been presented for illustrative purposes.

62F10 Point estimation
62E20 Asymptotic distribution theory in statistics
62N05 Reliability and life testing
62N02 Estimation in survival analysis and censored data
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[1] Awad, A.M.; Azzam, M.M.; Hamadan, M.A., Some inference results in \(P(Y < X)\) in the bivariate exponential model, Communications in statistics. theory and methods, 10, 2515-2524, (1981)
[2] Badar, M.G.; Priest, A.M., Statistical aspects of fiber and bundle strength in hybrid composites, (), 1129-1136
[3] Cheng, R.C.H.; Evans, B.E.; Iles, T.C., Embedded models in nonlinear regression, Journal of the royal statistical society, 54, 877-888, (1992) · Zbl 0774.62063
[4] Church, J.D.; Harris, B., The estimation of reliability from stress strength relationships, Technometrics, 12, 49-54, (1970) · Zbl 0195.20001
[5] Downtown, F., The estimation of \(P(X > Y)\) in the normal case, Technometrics, 15, 551-558, (1973) · Zbl 0262.62016
[6] Govidarajulu, Z., Two sided confidence limits for \(P(X > Y)\) based on normal samples of \(X\) and \(Y\), Sankhya B, 29, 35-40, (1967)
[7] Kotz, S.; Lumelskii, Y.; Pensky, M., The stress-strength model and its generalizations, (2003), World Scientific Press Singapore · Zbl 1017.62100
[8] Kundu, D.; Gupta, R.D., Estimation of \(P(Y < X)\) for the generalized exponential distribution, Metrika, 61, 3, 291-308, (2005) · Zbl 1079.62032
[9] Kundu, D.; Gupta, R.D., Estimation of \(P(Y < X)\) for Weibull distribution, IEEE transactions on reliability, 55, 2, 270-280, (2006)
[10] McCool, J.I., Inference on \(P(Y < X)\) in the Weibull case, Communications in statistics. simulation and computations, 20, 129-148, (1991) · Zbl 0850.62310
[11] Mudholkar, G.S.; Srivastava, D.K.; Kollia, G.D., A generalization of the Weibull distribution with application to the analysis of survival data, Journal of the American statistical association, 91, 1575-1583, (1996) · Zbl 0881.62017
[12] Owen, D.B.; Craswell, K.J.; Hanson, D.L., Non-parametric upper confidence bounds for \(P(Y < X)\) and confidence limits for \(P(Y < X)\) when \(X\) and \(Y\) are normal, Journal of the American statistical association, 59, 906-924, (1977) · Zbl 0127.10504
[13] Raqab, M.Z.; Kundu, D., Comparison of different estimators of \(P [Y < X]\) for a scaled burr type \(X\) distribution, Communications in statistics. simulation and computation, 34, 2, 465-483, (2005) · Zbl 1065.62172
[14] Raqab, M.Z.; Madi, M.T.; Kundu, D., Estimation of \(P(Y < X)\) for the 3-parameter generalized exponential distribution, Communications in statistics. theory and methods, 37, 18, 2854-2864, (2008) · Zbl 1292.62041
[15] Shao, Q.; Ip, W.; Wong, H., Determination of embedded distributions, Computational statistics & data analysis, 46, 317-334, (2004) · Zbl 1429.62088
[16] Smith, R.L., Maximum likelihood estimation in a class of non-regular cases, Biometrika, 72, 67-90, (1985) · Zbl 0583.62026
[17] Surles, J.G.; Padgett, W.J., Inference for \(P(Y < X)\) in the burr type X model, Journal of applied statistical sciences, 7, 225-238, (1998) · Zbl 0911.62092
[18] Surles, J.G.; Padgett, W.J., Inference for reliability and stress – strength for a scaled burr-type X distribution, Lifetime data analysis, 7, 187-200, (2001) · Zbl 0984.62082
[19] Woodward, W.A.; Kelley, G.D., Minimum variance unbiased estimation of \(P(Y < X)\) in the normal case, Technometrics, 19, 95-98, (1977) · Zbl 0352.62036
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