Errors of misclassification in discriminant analysis.

Abstract

This dissertation considers the estimation of the chance of misclassification when a new observation is assigned to one of two or more populations. (1) We have investigated by simulation, the effect on the chance of misclassification of replacing the usual unbiased estimates of $\mu\sb{i}(i=1,2)$ and $\Sigma\sp{-1}$, by more general, James-Stein estimators. (2) We have compared the optimum quadratic discriminant function with Anderson and Bahadur's linear function, in the case of different covariance matrices. (3) We have derived the large sample variances and covariances of Anderson's statistics when the hold-out method is used for estimating the chance of misclassification. Usually the correlations between successive values of Anderson's statistic are ignored and a simple binomial formula is used for obtaining a confidence interval for the true chance of misclassification. We include the correlations and derive a modified formula and study it by simulation. (4) We have extended Hills' (1966) result for two populations to more than two populations, in the hold-out method, when the classification rule is based on "scores". (5) The exact chance of misclassification can be expressed as a multiple integral of a multivariate normal distribution. Schervish (1981) expanded this integral but this form is not useful to a practitioner. We have approximated this integral by a Bonferroni type inequality, investigated its accuracy by simulation and have suggested an ad hoc correction to improve it. (6) We also approximated this integral by using an equicorrelated multivariate normal distribution as an approximation and used tables provided by Milton (1963) and by Gibbons, Olkin and Sobel (1977). The formula derived by us is very satisfactory and gives results close to the true values. (7) We compared Hudlet and Johnson's method of using an unweighted "Between groups" matrix with the standard weighted matrix, for obtaining canonical variables as discriminators. The standard method is convenient and better. (8) No useful measures of distance and canonical correlations are available in the literature for two or more than two populations, when the covariance matrices are unequal. We have proposed such measures in this dissertation.