New Developments on the Non-Central Chi-Squared and Beta Distributions

TL;DR

This paper introduces new formulas for the moments of the Non-central Chi-Squared and Beta distributions, using innovative approaches involving mixture representations and factorial expansions, validated through simulations.

Contribution

It presents novel moment formulas for these distributions, improving upon existing methods with new mathematical approaches and validation techniques.

Findings

New formulas for moments are derived using mixture and factorial expansion methods.

Simulation results validate the accuracy of the new formulas.

The new formulas offer advantages over previous methods in computational efficiency or accuracy.

Abstract

New formulas for the moments about zero of the Non-central Chi-Squared and the Non-central Beta distributions are achieved by means of novel approaches. The mixture representation of the former model and a new expansion of the ascending factorial of a binomial are the main ingredients of the first approach, whereas the second one hinges on an interesting relationship of conditional independence and a simple conditional density of the latter model. Then, a simulation study is carried out in order to pursue a twofold purpose: providing numerical validations of the derived moment formulas on one side and discussing the advantages of the new formulas over the existing ones on the other.