On Kolmogorov asymptotics of estimators of the misclassification error rate in linear discriminant analysis

We provide a fundamental theorem that can be used in conjunction with Kolmogorov asymptotic conditions to derive the first moments of well-known estimators of the actual error rate in linear discriminant analysis of a multivariate Gaussian model under the assumption of a common known covariance matrix. The estimators studied in this paper are plug-in and smoothed resubstitution error estimators, both of which have not been studied before under Kolmogorov asymptotic conditions. As a result of this work, we present an optimal smoothing parameter that makes the smoothed resubstitution an unbiased estimator of the true error. For the sake of completeness, we further show how to utilize the presented fundamental theorem to achieve several previously reported results, namely the first moment of the resubstitution estimator and the actual error rate. We provide numerical examples to show the accuracy of the succeeding finite sample approximations in situations where the number of dimensions is comparable or even larger than the sample size.