On Error Estimation for the Partitioning Classification Rule

Abstract

The resubstitution and the deleted error estimates for the partitioning classification rule from a sample (X ₁; Y ₁),...,(X _n, Y _n) are studied. The random part of the resubstitution estimate is shown to be small for arbitrary partition and for any distribution of (X, Y ). If we assume that X has a density f and the partitions consist of rectangles, then the difference between the expected value of the estimate and the Bayes error restricted to the partition is less than a constant times $1/\sqrt n$ . The main result of the paper is that, under the same conditions, the deleted estimate is asymptotically normal.