Bound on the Classification Error for Discriminating Between Multivariate Populations with Specified Means and Covariance Matrices

Abstract

Let $\mathscr{F}_1, \mathscr{F}_2$ be two families of $p$-variate distribution functions with specified means $\mathbf{\mu}_i (i = 1,2)$ and nonsingular covariance matrices $\Sigma_i$, and let $\pi_i$ be the prior probability assigned to $\mathscr{F}_i$ for $i = 1, 2$. The objective is to discriminate whether an observation $\mathbf{x}$ is from a distribution $F_1 \in \mathscr{F}_1$ or $F_2 \in \mathscr{F}_2$. Given a pair $F = (F_1, F_2)$ the error probability for classification rule $\phi$ is denoted by $e(\phi, F)$. In this paper the values of $\sup_F \inf_\phi e(\phi, F)$ and $\inf_\phi \sup_F e(\phi, F)$ are found and conditions for the existence of a saddle point of $e(\phi, F)$ are given. Also a saddle point is found when it exists. When $\phi$ is restricted to linear classification rules the same problems are considered. The mathematical programming method for finding a saddle point is also outlined.