Computer Algebra and Algorithms for Unbiased Moment Estimation of Arbitrary Order

Area of Science:

• Statistics
• Computational Statistics
• Statistical Software Development

Background:

• Unbiased estimators for lower-order central moments (e.g., variance) are common.
• Deriving unbiased estimators for higher-order moments involves complex mathematics and combinatorics.
• Existing methods struggle with higher-order moment calculations, including powers and products.

Purpose of the Study:

• To develop a generalizable software algorithm for computing unbiased estimators of arbitrary order central moments.
• To provide a practical tool for obtaining unbiased estimators up to the 6th order.
• To extend the methodology for pooled estimates of higher central moments from multiple populations.

Main Methods:

• Development of a novel software algorithm to compute unbiased estimators.
• Implementation of the algorithm in an R package named Umoments.
• Calculation of one- and two-sample unbiased estimates, including intermediate results.

Main Results:

• The algorithm successfully computes unbiased estimators for central moments up to the 6th order.
• The Umoments package provides efficient calculation of these estimators.
• The method supports the computation of pooled estimates for multiple populations, useful for two-sample tests.

Conclusions:

• The developed algorithm and Umoments package significantly simplify the computation of higher-order unbiased central moment estimators.
• This tool enhances statistical analysis capabilities, particularly for complex moment calculations and comparative studies.
• The software facilitates advanced statistical inference by providing reliable higher-order moment estimates.