Maximum Likelihood Estimation for Totally Positive Log-Concave Densities

TL;DR

This paper investigates nonparametric maximum likelihood estimation for multivariate distributions with positive dependence, proving existence and uniqueness of the MLE under certain conditions, and providing algorithms for computation.

Contribution

It establishes the existence and uniqueness of the MLE for log-supermodular and LLC distributions, and proves the MLE is an exponential of a tent function in specific cases.

Findings

MLE exists almost surely and is unique with high probability for n ≥ 3

MLE is the exponential of a tent function in certain cases

Provides a conditional gradient algorithm for computing the MLE

Abstract

We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP$_2$) distributions and log-$L^\#$-concave (LLC) distributions. In both cases we also assume log-concavity in order to ensure boundedness of the likelihood function. Given $n$ independent and identically distributed random vectors in $\mathbb R^d$ from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when $n\geq 3$. This holds independently of the ambient dimension $d$. We conjecture that the MLE is always the exponential of a tent function. We prove this result for samples in $\{0,1\}^d$ or in $\mathbb{R}^2$ under MTP$_2$, and for samples in $\mathbb{Q}^d$ under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.