Distribution Theory

This chapter provides results on sampling moments and distributions of error rates for classification rules specified by general discriminants, including the true error rate and the resubstitution, leave-one-out, cross-validation, and bootstrap error estimators. It also addresses the issue of separate sampling, side by side with mixture sampling, in a unified framework. The sampling issue creates problems for classifier design and error estimation. An example from Esfahani and Dougherty (2014a) illustrates the discrepancy in expected linear discriminant analysis (LDA) classification error between the two sampling mechanisms. The chapter discusses the sample-based discriminants with an eye toward differentiating between mixture and separate sampling. Finally, it describes the higher-order moments of the true error rate and of the resubstitution and leave-one-out estimated error rates, as well as the sampling distributions of resubstitution and leave-one-out.