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Estimation of \(\Pr [X>Y]\) for gamma distributions. (English) Zbl 0609.62129
The problem of estimating \(P=\Pr [X>Y]\), where X and Y have distribution functions F and G, has been considered by several authors in a parametric as well as nonparametric setup. The practical importance of the problem is highlighted by the fact that the probability signifies the reliability of a component of strength X which is subjected to a stress Y.
In the present paper, we are interested in the UMVU estimator of P when X and Y have independent gamma distributions with not necessarily equal scale and shape parameters. The UMVU estimator is derived in Section 2 based on uncensored samples from the distributions. The mean squared error of this estimator is compared by simulation (Section 3) with those of the ML estimator, and the UMVU estimator under a distribution-free framework. For the special case of exponential distributions, the UMVU estimator based on type-II censored samples is discussed in Section 4.

MSC:
62N05 Reliability and life testing
62F10 Point estimation
62G05 Nonparametric estimation
65C99 Probabilistic methods, stochastic differential equations
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