Decompositions for Sum-of-Power Statistics and the Sample Central Moments

TL;DR

This paper introduces useful decompositions of sum-of-power statistics, enabling efficient computation of sample moments like mean, variance, skewness, and kurtosis, especially for pooled samples and subgroups, with practical R implementations.

Contribution

It provides novel decompositions for sum-of-power statistics that facilitate calculating sample moments for pooled and subgroup data, improving computational efficiency.

Findings

Decompositions for sums of squares, cubes, and quads are derived.

Implemented in an accessible R function for practical use.

Enhances computation of sample moments in complex data structures.

Abstract

We give some useful decompositions of sum-of-powers statistics, leading to decompositions for the sample mean, sample variance, sample skewness and sample kurtosis. We solve two related problems: computing these sample moments for a pooled sample composed of subgroups with known moments; and computing these sample moments for a subgroup using known moments from other subgroups and the overall pooled sample. Each task is accomplished via decompositions of the sums-of-squares, sums-of-cubes and sums-of-quads from which the sample central moments (up to fourth order) are formed. We give decomposition results and we implement these in a user-friendly R function.