On the Posterior Probability estimate of the error rate of nonparametric classification rules

We address the problem of estimating the error probability of nonparametric classification rules. Instead of the well known counting-type estimators we propose a so called posterior probability estimator, which plugs a nonparametric estimate of the a posteriori probabilities into an algebraic expression of the error probability. We explore the properties of the plug-in estimator. Unlike the standard estimators, the variance of our estimator is shown to have some remarkable distribution-free properties for the k-nearest neighbor, kernel and histogram roles. We pay special attention of histogram classification rules, and show the consistency of the estimate in this case. Investigating the bias of the estimate we also obtain rate-of-convergence results under mild conditions on the distribution.