Effect of a shrinkage estimator on the linear discriminant function

The effect of a shrinkage estimator in linear discriminant analysis is examined based on the mean square error of the discriminant coefficients and the actual error rates. An asymptotic mean square error of discriminant coefficients with a shrinkage estimator is derived under multivariate normal populations. Asymptotic expansions of the expected actual error rates are also derived, using the perturbation method for the eigenvalues and eigenvectors of a random matrix. Some numerical experiments are made to examine the efficiency of a shrinkage estimator.