Computer Algebra and Algorithms for Unbiased Moment Estimation of Arbitrary Order.

Abstract

While unbiased central moment estimators of lower orders (such as a sample variance) are easily obtainable and often used in practice, derivation of unbiased estimators of higher orders might be more challenging due to long math and tricky combinatorics. Moreover, higher orders necessitate calculation of estimators of powers and products that also amount to these orders. We develop a software algorithm that allows the user to obtain unbiased estimators of an arbitrary order and provide results up to the 6th order, including powers and products of lower orders. The method also extends to finding pooled estimates of higher central moments of several different populations (e.g. for two-sample tests). We introduce an R package Umoments that calculates one- and two-sample estimates and generates intermediate results used to obtain these estimators.