Probability Inequalities and Errors in Classification

Abstract

Let $X$ and $Y$ be two $p \times 1$ random vectors distributed according to a normal distribution with respective mean vectors $\mu$ and $a\mu$ and covariance matrix $\begin{pmatrix}I_p & \rho I_p \\ \rho I_p & I_p\end{pmatrix}.$ Let $S$ be a random $p \times p$ matrix distributed as the Wishart distribution $W_p(I_p, r)$, independently of $X$ and $Y$. For fixed $a, \rho$, and $c$, some sufficient conditions are obtained for which $P\lbrack X'Y < c\rbrack$ and $P\lbrack X'S^{-1}Y < c\rbrack$ increase with $\mu'\mu$. These results are used to show a monotonicity property of the probabilities of correct classification of a class of rules for classifying an observation into one of two normal distributions. For the classification problem, some estimates of the probability of correct classification of the minimum distance rule are studied.