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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.

MSC:
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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