Gaussian Distribution Theory: Univariate Case

This chapter presents a survey of salient results in the distributional study of error rates of Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA) under an assumption of Gaussian populations. It concentrates on results for the separate-sampling population-specific error rates. The chapter presents results for the first moment of the true error and several error estimators for the separate-sampling case. An exact expression for the second-order moments of the true error is also given. The chapter also displays the RMS of the separate-sampling resubstitution and leave-one-out estimators of the overall error rate. It provides exact expressions for the marginal sampling distributions of the classical mixture-sampling resubstitution and leave-one-out estimators and discusses the joint sampling distribution and density of each of these error estimators and the true error in the univariate Gaussian case, and some of its applications, using results published in Zollanvari et al.