On the Laplace transform of some quadratic forms and the exact distribution of the sample variance from a gamma or uniform parent distribution

TL;DR

This paper derives explicit formulas for the distribution of the sample variance when the parent distribution is gamma or uniform, using Laplace transforms and integral representations, aiding statistical testing.

Contribution

It introduces new integral and Fourier series representations for the sample variance distribution from gamma and uniform distributions, extending previous results.

Findings

Derived a univariate integral for the CDF of the sample variance from gamma variables.

Obtained Fourier series for the density and distribution function of the sample variance from uniform variables.

Provided applications to statistical tests for scale parameters.

Abstract

From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for the cumulative distribution function of the sample variance of i.i.d. random variables with a gamma density, supplementing former formulas of the author. Furthermore, from the above Laplace transform Fourier series are obtained for the density and the distribution function of the sample variance of i.i.d. random variables with a uniform distribution. This distribution can be applied e.g. to a statistical test for a scale parameter.