# Discriminant analysis in small and large dimensions

## TL;DR

This paper analyzes the distribution and performance of linear discriminant analysis in both small and high-dimensional settings, providing new asymptotic results and efficient error rate computations.

## Contribution

It introduces a stochastic representation for discriminant coefficients and derives their asymptotic distribution in high dimensions, enhancing understanding of LDA's behavior.

## Key findings

- Asymptotic distribution of discriminant coefficients derived
- Efficient computation of error rates established
- Comparison with optimal error rates in known-parameter scenarios

## Abstract

We study the distributional properties of the linear discriminant function under the assumption of normality by comparing two groups with the same covariance matrix but different mean vectors. A stochastic representation for the discriminant function coefficients is derived which is then used to obtain their asymptotic distribution under the high-dimensional asymptotic regime. We investigate the performance of the classification analysis based on the discriminant function in both small and large dimensions. A stochastic representation is established which allows to compute the error rate in an efficient way. We further compare the calculated error rate with the optimal one obtained under the assumption that the covariance matrix and the two mean vectors are known. Finally, we present an analytical expression of the error rate calculated in the high-dimensional asymptotic regime. The finite-sample properties of the derived theoretical results are assessed via an extensive Monte Carlo study.