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

Carlo Orsi

doi:10.17713/ajs.v51i1.1106

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.

Highlights

In the present paper new expressions for the r-th moment about zero of the Non-central Chi-Squared and the Non-central Beta distributions are obtained
An approach to the analysis of the Doubly Non-central Beta distribution, i.e. the most general non-central extension of the Beta one, is made explicit. This approach rests on an interesting relationship of conditional independence and a suitable conditional density of such a model
These findings provide an analytical tool-kit that paves the way towards obtaining a surprisingly simple solution to the issue of assessing the moments of such a distribution

Summary

Introduction

In the present paper new expressions for the r-th moment about zero of the Non-central Chi-Squared and the Non-central Beta distributions are obtained. This approach rests on an interesting relationship of conditional independence and a suitable conditional density of such a model. These findings provide an analytical tool-kit that paves the way towards obtaining a surprisingly simple solution to the issue of assessing the moments of such a distribution. The whole analysis is performed by using the statistical software R

Preliminaries

On the moments of the non-central chi-squared distribution

On the moments of the non-central Beta distributions

Simulation results

Concluding remarks