Maximum Likelihood Estimation for Totally Positive Log-Concave Densities

We study nonparametric density estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely multivariate totally positive distributions of order 2 (MTP$_2$, a.k.a. log-supermodular) and the subclass of log-$L^\natural$-concave (LLC) distributions. In both cases we impose the additional assumption of log-concavity in order to ensure boundedness of the likelihood function. Given $n$ independent and identically distributed random vectors from a $d$-dimensional MTP$_2$ distribution (LLC distribution, respectively), we show that the maximum likelihood estimator (MLE) exists and is unique with probability one when $n\geq 3$ ($n\geq 2$, respectively), independent of the number $d$ of variables. The logarithm of the MLE is a tent function in the binary setting and in $\mathbb{R}^2$ under MTP$_2$ and in the rational setting under LLC. We provide a conditional gradient algorithm for computing it, and we conjecture that the same convex program also yields the MLE in the remaining cases.