Maximum likelihood estimation for totally positive log‐concave densities

Elina Robeva, Bernd Sturmfels, Ngoc Tran and Caroline Uhler

Scandinavian Journal of Statistics, 2021, vol. 48, issue 3, 817-844

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 ℝ2 under MTP2, and for samples in ℚd under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.

Date: 2021
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