Maximum likelihood estimation for totally positive log-concave densities

Abstract

We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP2) 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 from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when n≥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 (Formula presented.) under MTP2, and for samples in (Formula presented.) under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.