Mean Square Errors of the Best Unbiased and Maximum Likelihood Estimators of Tail Probabilities in Normal Distributions

Formulas for the variance of the uniformly minimum variance unbiased (UMVU) estimator, and of the mean square error (MSE) of the maximum likelihood (ML) estimator, of tail probabilities of normal distributions are derived. Relative efficiency values of these estimators of samples of size 6, 12, 18 and tail probabilities .025, .05, .10, .2, .3, .4, .5 are presented. The bias of the ML estimator for these cases is also given. The formulas developed involve incomplete moments of non-central t-distributions and expectations of standard bivariate normal c.d.f., with respect to chi-square random variables.