# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] DOI: 10.1002/nav.3800280304 · Zbl 0462.62079 · doi:10.1002/nav.3800280304 [2] DOI: 10.1007/BF01893574 · Zbl 0436.62029 · doi:10.1007/BF01893574 [3] DOI: 10.1080/03610928008827882 · Zbl 0437.62035 · doi:10.1080/03610928008827882 [4] Birnbaum, Z.W. On a use of the Mann-Whitney statistic. Proceedings of the Third Berkeley Symposium. Vol. 1, pp.13–17. · Zbl 0071.35504 [5] DOI: 10.1002/nav.3800220104 · Zbl 0339.62071 · doi:10.1002/nav.3800220104 [6] Chandra S., South African Statistical Journal 11 pp 149– (1977) [7] DOI: 10.1214/aoms/1177728793 · Zbl 0056.38203 · doi:10.1214/aoms/1177728793 [8] DOI: 10.1007/BF02911637 · Zbl 0176.48606 · doi:10.1007/BF02911637 [9] DOI: 10.2307/2283110 · Zbl 0127.10504 · doi:10.2307/2283110 [10] DOI: 10.2307/3314627 · Zbl 0539.62036 · doi:10.2307/3314627 [11] DOI: 10.2307/1267617 · Zbl 0292.62021 · doi:10.2307/1267617
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.