Parameter Estimation For Multivariate Generalized Gaussian Distributions

TL;DR

This paper proves the existence and uniqueness of the maximum likelihood estimator for the scatter matrix of multivariate generalized Gaussian distributions and proposes an algorithm for its computation, demonstrating good performance on synthetic and real data.

Contribution

It establishes the theoretical existence and uniqueness of the MLE for MGGD parameters and introduces a Newton-Raphson based estimation algorithm.

Findings

MLE of the scatter matrix exists and is unique up to a scalar for shape parameter in (0,1)

The proposed algorithm converges efficiently in experiments

Parameters can be estimated accurately using MLE with good performance

Abstract

Due to its heavy-tailed and fully parametric form, the multivariate generalized Gaussian distribution (MGGD) has been receiving much attention for modeling extreme events in signal and image processing applications. Considering the estimation issue of the MGGD parameters, the main contribution of this paper is to prove that the maximum likelihood estimator (MLE) of the scatter matrix exists and is unique up to a scalar factor, for a given shape parameter \beta\in(0,1). Moreover, an estimation algorithm based on a Newton-Raphson recursion is proposed for computing the MLE of MGGD parameters. Various experiments conducted on synthetic and real data are presented to illustrate the theoretical derivations in terms of number of iterations and number of samples for different values of the shape parameter. The main conclusion of this work is that the parameters of MGGDs can be estimated using the maximum likelihood principle with good performance.