Progress on a Conjecture Regarding the Triangular Distribution

TL;DR

This paper proves a conjecture that maximum likelihood estimation for the mode of a triangular distribution can be efficiently performed with a constant number of likelihood evaluations, regardless of sample size.

Contribution

The paper provides rigorous proofs confirming that the conjectured constant evaluation method is valid for large samples, supported by graphical and numerical evidence.

Findings

Validation of the conjecture through two theorems

Efficient estimation method with constant evaluations

Supporting graphical and numerical results

Abstract

Triangular distributions are a well-known class of distributions that are often used as an elementary example of a probability model. Maximum likelihood estimation of the mode parameter of the triangular distribution over the unit interval can be performed via an order statistics-based method. It had been conjectured that such a method can be conducted using only a constant number of likelihood function evaluations, on average, as the sample size becomes large. We prove two theorems that validate this conjecture. Graphical and numerical results are presented to supplement our proofs.