Asymptotic expansion and estimation of EPMC for linear classification rules in high dimension

The problem of classifying a new observation vector into one of the two known groups distributed as multivariate normal with common covariance matrix is considered. In this paper, we handle the situation that the dimension, pp, of the observation vectors is less than the total number, NN, of observation vectors from the two groups, but both pp and NN tend to infinity with the same order. Since the inverse of the sample covariance matrix is close to an ill condition in this situation, it may be better to replace it with the inverse of the ridge-type estimator of the covariance matrix in the linear discriminant analysis (LDA). The resulting rule is called the ridge-type linear discriminant analysis (RLDA). The second-order expansion of the expected probability of misclassification (EPMC) for RLDA is derived, and the second-order unbiased estimator of EMPC is given. These results not only provide the corresponding conclusions for LDA, but also clarify the condition that RLDA improves on LDA in terms of EPMC. Finally, the performances of the second-order approximation and the unbiased estimator are investigated by simulation.