Exact Distribution of the Sample Variance from a Gamma Parent Distribution

Several representations of the exact cdf of the sum of squares of n independent identically gamma–distributed random variables Xi are given, in particular by a series of gamma distribution functions. Using a characterization of the gamma distribution by Laha, an expansion of the exact distribution of the sample variance is derived by a Taylor series approach with the former distribution as its leading term. In particular for integer orders α some further series are provided, including a convex combination of gamma distributions for α = 1 and nearly of this type for α > 1. Furthermore, some representations of the distribution of the angle Φ between (X1, ...,Xn) and (1, ...,1) are given by orthogonal series. All these series are based on the same sequence of easily computed moments of cos(Φ). AMS 2000 Subject Classifications: 62E15, 62H10