ASYMPTOTIC EVALUATION OF THE PROBABILITIES OF MISCLASSIFICATION BY LINEAR DISCRIMINANT FUNCTIONS

Abstract

This chapter presents an asymptotic evaluation of the probabilities of misclassification by linear discriminant functions. The problem of classifying an observation into one of two multivariate normal populations with a common covariance matrix might be called the classical classification problem. Fisher's linear discriminant function serves as a criterion when samples are used to estimate the parameters of the two distributions. When the parameters are unknown and there is available a sample from each population, the mean of each population is estimated by the mean of the respective sample and the common covariance matrix of the populations is estimated by using deviations from the respective means in the two samples. The classification function W, proposed by T. W. Anderson, is obtained by replacing the parameters in the linear function resulting from the Neyman–Pearson Fundamental Lemma by the estimates; the substitution for parameters has been called plugging-in estimates. The Studentized W statistic is W less the estimate of its limiting mean divided by the estimate of its limiting standard deviation. If a statistician wants to set his cut-off point to achieve a specified probability of misclassification, he can use this Studentized W.