Minimum variance estimators for misclassification probabilities in discriminant analysis

Let [alpha](n1, n2) be the probability of classifying an observation from population [Pi]1 into population [Pi]2 using Fisher's linear discriminant function based on samples of size n1 and n2. A standard estimator of [alpha], denoted by T1, is the proportion of observations in the first sample misclassified by the discriminant function. A modification of T1, denoted by T2, is obtained by eliminating the observation being classified from the calculation of the discriminant function. The UMVU estimators, and , of ET1 = [tau]1(n1, n2) and ET2 = [tau]2(n1, n2) = [alpha](n1 - 1, n2) are derived for the case when the populations have multivariate normal distributions with common dispersion matrix. It is shown that and are nonincreasing functions of D2, the Mahalanobis sample distance. This result is used to derive the sampling distributions and moments of and . It is also shown that [alpha] is a decreasing function of [Delta]2 = ([mu]1 - [mu]2)'[Sigma]-1([mu]1 - [mu]2). Hence, by truncating and (or any estimator) at the value of [alpha] for [Sigma] = 0, new estimators are obtained which, for all samples, are as close or closer to [alpha].