Moments estimators and omnibus chi-square tests for some usual probability laws

TL;DR

This paper develops a method using the functional empirical process to derive joint distribution laws and omnibus chi-square tests for moment estimators across several probability laws, with practical simulation results.

Contribution

It introduces a practical approach to derive asymptotic distributions and chi-square tests for moment estimators using the functional empirical process, applied to four common distributions.

Findings

Omnibus chi-square tests perform well with sample sizes around fifty.

The method allows algebraic derivation of joint distribution laws for moment estimators.

Simulations confirm the effectiveness of the proposed tests.

Abstract

For many probability laws, in parametric models, the estimation of the parameters can be done in the frame of the maximum likelihood method, or in the frame of moment estimation methods, or by using the plug-in method, etc. Usually, for estimating more than one parameter, the same frame is used. We focus on the moment estimation method in this paper. We use the instrumental tool of the functional empirical process (fep) in Lo (2016) to show how it is practical to derive, almost algebraically, the joint distribution Gaussian law and to derive omnibus chi-square asymptotic laws from it. We choose four distributions to illustrate the method (Gamma law, beta law, Uniform law and Fisher law) and completely describe the asymptotic laws of the moment estimators whenever possible. Simulations studies are performed to investigate for each case the smallest sizes for which the obtained statistical tests are recommendable. Generally, the omnibus chi-square test proposed here work fine with sample sizes around fifty