Accurate distribution of X^{T}X with singular, idempotent variance-covariance matrix

TL;DR

This paper refines the distribution approximation of X^{T}X for sample statistics with idempotent covariance, incorporating (1/n) corrections to improve accuracy for small samples.

Contribution

It extends the known chi-squared approximation of X^{T}X to include (1/n) corrections, enhancing accuracy for small sample sizes.

Findings

Improved approximation accuracy for small samples.

Extension of chi-squared distribution with correction terms.

Broader applicability of the distribution approximation.

Abstract

Assume that X is a set of sample statistics which follow a special case Central Limit Theorem, namely: as the sample size n increases the corresponding distribution becomes multivariate Normal with the mean (of each X) equal to zero and with an idempotent variance-covariance matrix V. It is well known that X^{T}X has (in the same limit), a chi-squared distribution with degrees of freedom equal to the trace of V. In this article we extend the above result to include the corresponding (1/n)-proportional corrections, making the new approximation substantially more accurate and extending its range of applicability to small-size samples.