Gaussian Distribution Theory: Multivariate Case

This chapter surveys results in the distributional theory of error rates of the Linear (LDA) and Quadratic Discriminant Analysis (QDA) classification rules under multivariate Gaussian populations, with the aim of obtaining exact or approximate (accurate) expressions for the moments of true and estimated error rates. It focuses on two approaches: small-sample and large sample methods. It examines small-sample methods based on statistical representations for the discriminants defined in the case of Gaussian populations and separate sampling. In this chapter, statistical representations are given for W(Sn0,n1, X) | Y = 0 and W(Sn0,n1, X1) | Y1 = 0 in the case of Anderson's and John's LDA discriminants, as well as the general QDA discriminant, under multivariate Gaussian populations and separate sampling. Finally, the chapter shows how the statistical representations can be useful for Monte-Carlo computation of the error moments and error distribution based on the discriminant distribution.