Quadratic discriminant functions with constraints on the covariance matrices: Some asymptotic results

In multivariate discrimination of two normal populations, the optimal classification procedure is based on the quadratic discriminant function. We investigate asymptotic properties of this function if the covariance matrices of the two populations are estimated under the following four models; (i) arbitrary covariance matrices; (ii) common principal components, that is, equality of the eigenvectors of both covariance matrices; (iii) proportional covariance matrices; and (iv) identical covariance matrices. It is shown that using a restricted model, provided that it is correct, often yields smaller asymptotic variances of discriminant function coefficients than the usual quadratic discriminant approach with no constraints on the covariance matrices. In particular, the proportional model appears to provide an attractive compromise between linear and ordinary quadratic discrimination.