On Mardia skewness and kurtosis of Soules basis matrices in ROM simulation

Abstract

Relying on an interesting connection between generalized Helmert–Ledermann matrices and Soules basis matrices, as well as a complete binary tree classification of the latter, a recursive determination of the Mardia skewness and kurtosis coefficients of Soules basis matrices is proposed. It is based on a convenient combinatorial representation of Soules basis matrices by restricted binary partitions of a natural number into two ordered summands with recursive repetition of this binary partitioning on the summands. The obtained simple composition and invariance formulas enable the recursive calculation of these skewness and kurtosis coefficients.