Recurrence relations for the moments of order statistics from a beta distribution

Abstract

Order statistics, Single and product moments,

Reduced Complexity Recursive Grassmannian Quantization

“…2 is determined by the maximum order statistics of D independent beta random variables. Utilizing the recurrence relation of [19], we can calculate the expected value μ…”

“…For small codebook size D, this bound is tighter than the bound of [5]; for m = 1 or D = 1 the bound is even achieved with equality. However, for large codebook sizes, calculating the recurrence relation of [19] becomes prohibitively complex; we then apply the asymptotically tight RVQ bound of [5].…”

Integrals of incomplete beta functions, with applications to order statistics, random walks and string enumeration

“…Calculating properties of the maximum of a set of independent random variables falls, of course, under the more general umbrella of determining their order statistics. In the case of beta random variables, methods are provided in Thomas andSamuel (2008)andAbdelkader (2010) for computing moments of order statistics in the case where the parameters of the beta distribution(s) are all integer-valued. However, the resulting equations are either recursive (Thomas and Samuel (2008)) or quite complicated nested sums of 'inclusion exclusion' type (Abdelkader (2010)).…”

“…In the case of beta random variables, methods are provided in Thomas andSamuel (2008)andAbdelkader (2010) for computing moments of order statistics in the case where the parameters of the beta distribution(s) are all integer-valued. However, the resulting equations are either recursive (Thomas and Samuel (2008)) or quite complicated nested sums of 'inclusion exclusion' type (Abdelkader (2010)). For example, in Thomas and Samuel (2008) (in which it is also assumed that the random variables are all identically distributed) the expectation of the maximum of two Beta(m, n) random variables is expressed as a linear combination of the first m + n moments of a single Beta(m, n).…”

Intriguing yet simple skewness: kurtosis relation in economic and demographic data distributions, pointing to preferential attachment processes

“…Balakrishnan et al (1988) have reviewed many recurrence relations and identities for several continuous distributions. Pertinently, in view of the following, let us point to Thomas and Samuel (2008) analysis of recurrence relations for the Beta distribution moments, -a distribution of wide application, both in its continuous or discrete form (Johnson et al, 1995;Martinez-Mekler et al, 2009;Ausloos and Cerqueti, 2016a) The skewness and kurtosis are officially the third and fourth moment, µ 3 and µ 4 respectively, of a distribution, where µ i is usually centered on the mean; µ 2 is of course the variance, sometimes called σ 2 . Mathematical statistics textbooks and software packages usually calculate the Fisher-Pearson coefficient of skewness and the kurtosis S = µ 3 / µ 3/2 2 (1)…”