Robust Nonparametric Discriminant Analysis Procedures

Abstract

In this study, a nonparametric discriminant analysis procedure that is less sensitive than traditional procedures to deviations from the usual assumptions is proposed. The procedure uses the projection pursuit methodology where the projection index is the two-group transvariation probability. Montanari (2004) proposed and used this projection index to measure group separation but allocated the new observation using simple Euclidean distances from projected centers. Our procedure employs a method of allocation based on the centrality of the new point measured using two versions of the transvariation probability: a symmetrized two-group transvariation and a smooth version of point-group transvariation. It is shown by simulation that the procedures proposed in this study provide lower misclassification error rates than classical procedures such as linear discriminant analysis and quadratic discriminant analysis and recent procedures like maximum depth and Montanari's transvariation-based classifiers under a variety of distributional settings. A different rank-based procedure for classification is considered where ranking is applied on classical classifiers as well as recently introduced classifiers such as the maximum L1 depth and quadratic discriminant function based on the minimum covariance determinant (MCD) estimates of the mean and covariance. An extensive simulation study shows that not only does the ranking method provide balance between misclassification error rates for each group but also yields lower total probabilities of misclassification and higher consistency of correct classification for heavy-tailed distributions. A theoretical evaluation of the influence function shows that this new procedure is robust against local infinitesimal contaminations.