MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Maximum Likelihood Estimation for Totally Positive Log-Concave Densities

Elina Robeva, Bernd Sturmfels, Ngoc Tran and Caroline Uhler


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 (MTP2, a.k.a. log-supermodular) and the subclass of log-L♮-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 MTP2 distribution (LLC distribution, respectively), we show that the maximum likelihood estimator (MLE) exists and is unique with probability one when n≥3 (n≥2, respectively), independent of the number d of variables. The logarithm of the MLE is a tent function in the binary setting and in R2 under MTP2 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.


Related publications

2021 Repository Open Access
Elina Robeva, Bernd Sturmfels, Ngoc Mai Tran and Caroline Uhler

Maximum likelihood estimation for totally positive log-concave densities

In: Scandinavian journal of statistics, 48 (2021) 3, pp. 817-844