The exact distribution of the sample variance from bounded continuous random variables

TL;DR

This paper derives an exact formula for the distribution of the sample variance for bounded continuous i.i.d. variables, using Fourier series and special functions, with extensions to quadratic forms of independent variables.

Contribution

It provides a novel integral representation of the sample variance distribution for bounded variables and generalizes the method to quadratic forms of independent variables.

Findings

Exact distribution expressed as a Fourier series integral

Coefficients simplified for polynomial and trigonometrical polynomial densities

Extension to quadratic forms with rank-1 perturbations

Abstract

For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial or a trigonometrical polynomial the coefficients of this series are simple finite terms containing only the error function, the exponential function and powers. In more general cases - e.g. for all beta densities - the coefficients are given by some series expansions. The method is generalized to positive semi-definite quadratic forms of bounded independent but not necessarily identically distributed random variables if the form matrix differs from a diagonal matrix D > 0 only by a matrix of rank 1