Generalized Consistent Error Estimator of Linear Discriminant Analysis

A classifier is epistemologically vacuous without an accurate estimate of its true error rate. In situations where the number of sample points is of the same order of magnitude as the dimension of observations, serious issues arise with respect to the performance of error estimators. In this paper, we place the problem of synthesizing an error rate estimator of a common linear classifier in an asymptotic setting in which the number of sample points is kept comparable in magnitude to the dimension of observations (double asymptotic). We construct a generalized consistent estimator of the true error rate for linear discriminant analysis in the multivariate Gaussian model under the assumption of a common covariance matrix. In other words, the estimator converges to true error rate in the double asymptotic sense. We employ simulations using both synthetic and real data to compare the performance of the new estimator to the classical consistent estimator of the true error (plug-in estimator) as well as other well-known estimators. We observe that the constructed estimator can outperform other estimators of the true error in many situations in terms of bias and root-mean-square (RMS) error.