Estimating a Polya frequency function_2

TL;DR

This paper develops a non-parametric maximum likelihood estimator for Polya frequency functions of order two, showing it can be computed via convex programming and achieves consistency under certain conditions.

Contribution

It introduces a convex programming approach for estimating PFF_2 densities and provides an iterative algorithm for practical computation.

Findings

Estimator is the solution to a convex optimization problem.

Algorithm similar to iterative convex minorant method is devised.

Estimator achieves Hellinger consistency when the true density is PFF_2.

Abstract

We consider the non-parametric maximum likelihood estimation in the class of Polya frequency functions of order two, viz. the densities with a concave logarithm. This is a subclass of unimodal densities and fairly rich in general. The NPMLE is shown to be the solution to a convex programming problem in the Euclidean space and an algorithm is devised similar to the iterative convex minorant algorithm by Jongbleod (1999). The estimator achieves Hellinger consistency when the true density is a PFF_2 itself.