The distribution of the square sum of Dirichlet random variables and a table with quantiles of Greenwood's statistic

TL;DR

This paper derives exact and series-based representations for the distribution of the square sum of Dirichlet random variables, including Greenwood's statistic, and provides detailed quantile tables for practical use.

Contribution

It introduces new integral and orthogonal series representations for the distribution, including a special case for Greenwood's statistic, with comprehensive quantile tables.

Findings

Exact distribution formulas for the square sum of Dirichlet variables.

Orthogonal series representations using Jacobi and Legendre polynomials.

Quantile tables for Greenwood's statistic from 10 to 100 squares.

Abstract

The exact distribution of the square sum of Dirichlet random variables is given by two different univariate integral representations. Alternatively, three representations by orthogonal series with Jacobi or Legendre polynomials are derived. As a special case the distribution of the square sum of spacings - also called Greenwood's statistic - is obtained. Nine quantiles of this statistic are tabulated with eight digits where the number of squares ranges from 10 to 100.