Nonparametrically consistent depth-based classifiers

We introduce a class of depth-based classification procedures that are of a nearest-neighbor nature. Depth, after symmetrization, indeed provides the center-outward ordering that is necessary and sufficient to define nearest neighbors. Like all their depth-based competitors, the resulting classifiers are affine-invariant, hence in particular are insensitive to unit changes. Unlike the former, however, the latter achieve Bayes consistency under virtually any absolutely continuous distributions - a concept we call nonparametric consistency, to stress the difference with the stronger universal consistency of the standard $k$NN classifiers. We investigate the finite-sample performances of the proposed classifiers through simulations and show that they outperform affine-invariant nearest-neighbor classifiers obtained through an obvious standardization construction. We illustrate the practical value of our classifiers on two real data examples. Finally, we shortly discuss the possible uses of our depth-based neighbors in other inference problems.