Entropy-based test for generalized Gaussian distributions

TL;DR

This paper introduces a non-parametric goodness-of-fit test for generalized Gaussian distributions using an entropy estimator, supported by theoretical proofs and numerical simulations.

Contribution

It provides the first proof of $L^2$ consistency for the $k$th nearest neighbor entropy estimator and develops a new entropy-based test for generalized Gaussian distributions.

Findings

The $k$th nearest neighbor entropy estimator is $L^2$ consistent.

The proposed test effectively distinguishes generalized Gaussian distributions.

Numerical studies validate the theoretical results.

Abstract

In this paper, we provide the proof of $L^2$ consistency for the $k$th nearest neighbour distance estimator of the Shannon entropy for an arbitrary fixed $k\geq 1.$ We construct the non-parametric test of goodness-of-fit for a class of introduced generalized multivariate Gaussian distributions based on a maximum entropy principle. The theoretical results are followed by numerical studies on simulated samples.