Nonparametrically consistent depth-based classifiers

TL;DR

This paper introduces a new class of depth-based classifiers that are affine-invariant and achieve nonparametric Bayes consistency, outperforming standard affine-invariant nearest-neighbor methods in simulations and real data applications.

Contribution

The paper proposes a novel depth-based classification method that guarantees nonparametric consistency under broad conditions, unlike existing depth-based classifiers.

Findings

Outperform standard affine-invariant nearest-neighbor classifiers in simulations.

Achieve nonparametric Bayes consistency under virtually any absolutely continuous distribution.

Demonstrate practical effectiveness on real data examples.

Abstract

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.