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

2008/10/09 by Thomas Royen, T. Royen, Royen, T.
Decision Sciences · Mathematics · Physics and Astronomy · #62E15 #62H10 #FOS: Mathematics #Probability and Risk Models #Scientific Research and Discoveries #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #msc:62E15 #msc:62H10 #stat.TH

paper · pdf · doi:10.48550/arxiv.0810.1572

20 pages

arxiv created 2008/10/09 · openalex publication_date 2008/10/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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

Citations

• Exact distribution of the sample variance from a gamma parent distribution
• Integral representations for convolutions of non-central multivariate gamma distributions
• On the Laplace transform of some quadratic forms and the exact distribution of the sample variance from a gamma or uniform parent distribution

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