Unsupervised linear discrimination using skewness

TL;DR

This paper investigates unsupervised methods for linear discrimination based on skewness, deriving their statistical properties and comparing their asymptotic behaviors through simulations.

Contribution

It introduces two new estimators for unsupervised linear discrimination using skewness and analyzes their asymptotic distributions.

Findings

All affine equivariant estimators have proportional asymptotic covariance matrices.

Simulations confirm theoretical results and finite-sample behaviors.

Comparison of estimators simplifies due to proportional covariance matrices.

Abstract

It is well-known that, in Gaussian two-group separation, the optimally discriminating projection direction can be estimated without any knowledge on the group labels. In this work, we \revision{gather} several such unsupervised estimators based on skewness and derive their limiting distributions. As one of our main results, we show that all affine equivariant estimators of the optimal direction have proportional asymptotic covariance matrices, making their comparison straightforward. Two of our four estimators are novel and two have been proposed already earlier. We use simulations to verify our results and to inspect the finite-sample behaviors of the estimators.