A comparison of several biased estimators for improving the expected error rate of the sample quadratic discriminant function

The sample quadratic discriminant function (QDF) has been shown by Marks and Dunn (1974) to be superior to the linear discriminant function for two normal populations with , provided the training sample sizesn 1and n 2, are sufficiently large. However, the performance of the QDF quickly deteriorates as the dimension p increases relative to the sample size n i i = l , 2 . The deterioration is principally due to poor estimates of the inverse of the covariance matrices, . One method of combating this problem is to apply biased estimators of the inverse of the covariance matrices. In this paper we contrast the performance of the QDF with respect to several biased estimators and one unbiased estimator of A shrinkage estimator proposed by Peck and Van Ness (1982) is found to yield superior performance over a wide range of configurations and training sample sizes.