Unbiased estimators of \(P(X<Y)\) and their variances in the case of uniform and two-parameter exponential distributions
The statistical estimation of the probability of linear inequalities is widely used in practice. In particular, in reliability theory the problem is to estimate the probability of failure-free operation of a compound technical system. Let us assume that the system breaks down if the ``stress'' \(X\) exceeds the ``strength'' \(Y\). In this case, the terms ``stress'' and ``strength'' are understood in a wider meaning. If we assume that \(X\) and \(Y\) are independent random variables, then the probability of failure-free operation is \[ {\mathbf P}(X<Y).\tag{1} \] We consider the two-parameter exponential distribution with unknown shift and known scale parameters as well as the uniform distribution on a segment with one end known. For these cases unbiased estimators for (1) are found. We also obtain unbiased estimators for the variances of these unbiased estimators.
scientific article; zbMATH DE number 1011669