Maximum Likelihood Estimation of Triangular and Polygonal Distributions

TL;DR

This paper introduces a new parametrization and MM algorithm for maximum likelihood estimation of triangular and polygonal distributions, demonstrating improved convergence and performance over existing methods.

Contribution

A novel parametrization enabling globally convergent MM algorithms for ML estimation of triangular and polygonal distributions, with demonstrated numerical advantages.

Findings

The new MM algorithm monotonically increases likelihoods.

The algorithm is globally convergent.

Numerical simulations show improved performance.

Abstract

Triangular distributions are a well-known class of distributions that are often used as elementary example of a probability model. In the past, enumeration and order statistic-based methods have been suggested for the maximum likelihood (ML) estimation of such distributions. A novel parametrization of triangular distributions is presented. The parametrization allows for the construction of an MM (minorization--maximization) algorithm for the ML estimation of triangular distributions. The algorithm is shown to both monotonically increase the likelihood evaluations, and be globally convergent. Using the parametrization is then applied to construct an MM algorithm for the ML estimation of polygonal distributions. This algorithm is shown to have the same numerical properties as that of the triangular distribution. Numerical simulation are provided to demonstrate the performances of the new algorithms against established enumeration and order statistics-based methods.