Small Sample Statistics for Classification Error Rates I: Error Rate Measurements

Abstract

Several methods (independent subsamples, leave-one-out, cross-validation, and bootstrapping) have been proposed for estimating the error rates of classifiers. The rationale behind the various estimators and the causes ofthe sometimes conflicting claims regarding their bias and precision are explored in this paper. The biases and variances of each of the estimators are examined empirically. Cross-validation, 10-fold or greater, seems to be the best approach, the other methods are biased, have poorer precision, or are inconsistent. (Though unbiased for linear discriminant classifiers, the 632b bootstrap estimator is biased for nearest neighbors classifiers, more so for single nearest neighbor than for three nearest neighbors. The 632b estimator is also biased for CART-style decision trees. Weiss' LOO* estimator is unbiased and has better precision than cross-validation for discriminant and nearest neighbors classifiers, but its lack of bias and improved precision for those classifiers do not carry over to decision trees for nominal attributes.