  We study the distributional properties of the linear discriminant function
under the assumption of normality by comparing two groups with the same
covariance matrix but different mean vectors. A stochastic representation for
the discriminant function coefficients is derived which is then used to obtain
their asymptotic distribution under the high-dimensional asymptotic regime. We
investigate the performance of the classification analysis based on the
discriminant function in both small and large dimensions. A stochastic
representation is established which allows to compute the error rate in an
efficient way. We further compare the calculated error rate with the optimal
one obtained under the assumption that the covariance matrix and the two mean
vectors are known. Finally, we present an analytical expression of the error
rate calculated in the high-dimensional asymptotic regime. The finite-sample
properties of the derived theoretical results are assessed via an extensive
Monte Carlo study.
