On some nonparametric classifiers based on distribution functions of multivariate ranks

Over the last two decades, multivariate sign and rank based methods have become popular in analysing multivariate data. In this paper, we propose a classification methodology based on the distribution of multivariate rank functions. The proposed method is fully nonparametric in nature. Initially, we consider a theoretical version of the classifier for K populations and show that it is equivalent to the Bayes rule for spherically symmetric distributions with a location shift. Then we present the empirical version of that and show that the apparent misclassification rate of the empirical version of the classifier converges asymptotically to the Bayes risk. We also present an affine invariant version of the classifier and its optimality for elliptically symmetric distributions. We illustrate the performance in comparison with some other depth based classifiers using simulated and real data sets.