On the maximization of likelihoods belonging to the exponential family using ideas related to the Levenberg-Marquardt approach

TL;DR

This paper explores adapting the Levenberg-Marquardt algorithm for maximizing likelihoods in the exponential family, demonstrating local convergence and effective application to compositional data.

Contribution

It introduces a novel adaptation of the Levenberg-Marquardt method for exponential family likelihood maximization, with theoretical convergence analysis and practical implementation insights.

Findings

Demonstrates local convergence of the adapted algorithm

Shows stability and efficiency in real and simulated data

Provides implementation strategies for penalization

Abstract

The Levenberg-Marquardt algorithm is a flexible iterative procedure used to solve non-linear least squares problems. In this work we study how a class of possible adaptations of this procedure can be used to solve maximum likelihood problems when the underlying distributions are in the exponential family. We formally demonstrate a local convergence property and we discuss a possible implementation of the penalization involved in this class of algorithms. Applications to real and simulated compositional data show the stability and efficiency of this approach.