Large-Sample Properties of Blind Estimation of the Linear Discriminant Using Projection Pursuit

TL;DR

This paper analyzes the asymptotic properties of blind estimation of the linear discriminant using projection pursuit, deriving central limit theorems and comparing efficiency with classical LDA.

Contribution

It provides the first asymptotic analysis of blind projection pursuit estimators for linear discriminants, including CLTs for various projection indices.

Findings

Projection pursuit estimators achieve LDA efficiency with well-separated groups.

Asymptotic variances are proportional to those of LDA.

Finite sample validity of asymptotic formulas is supported by simulations.

Abstract

We study the estimation of the linear discriminant with projection pursuit, a method that is blind in the sense that it does not use the class labels in the estimation. Our viewpoint is asymptotic and, as our main contribution, we derive central limit theorems for estimators based on three different projection indices, skewness, kurtosis and their convex combination. The results show that in each case the limiting covariance matrix is proportional to that of linear discriminant analysis (LDA), an unblind estimator of the discriminant. An extensive comparative study between the asymptotic variances reveals that projection pursuit is able to achieve efficiency equal to LDA when the groups are arbitrarily well-separated and their sizes are reasonably balanced. We conclude with a real data example and a simulation study investigating the validity of the obtained asymptotic formulas for finite samples.