Constrained linear discriminant rule for classification of two groups via the Studentized classification statistic W for large dimension

Abstract

This paper is concerned with $2$-group linear discriminant analysis for multivariate normal populations with unknown mean vectors and unknown common covariance matrix for the case in which the sample sizes $N_1$, $N_2$ and the dimension $p$ are large. We give Studentized version of the $W$ statistic under the high-dimensional asymptotic framework A1 that $N_1$, $N_2$, and $p$ tend to infinity together under the condition that $p/(N_1+N_2-2)$ converges to a constant in $(0,1)$, and $N_1/N_2$ converges to a constant in $(0,\infty )$. Asymptotic expansion of the distribution for the conditional probability of misclassification (CPMC) of the Studentized $W$ is derived under A1. By using this asymptotic expansion, we give the cut-off point such that the one of two CPMCs is less than the presetting value. Such the constrained discriminant rule is studied by Anderson (1973) and McLachlan (1977). Simulation result reveals that the proposed method is more accurate than McLachlan (1977)’s method for the case in which $p$ is relatively large.