Statistica Sinica 28 (2018), 2697-2712
Abstract: We define a new class of log-concave distributions on the discrete lattice 𝕫 𝒹, and study its properties. We show how to compute the maximum likelihood estimator of this class of probability mass functions from an independent and identically distributed sample, and establish consistency of the estimator, even if the class has been incorrectly specified. For finite sample sizes, in our simulations, the proposed estimator outperforms a purely nonparametric approach (the empirical distribution), but is able to remain comparable to the correct parametric approach. Notably, the new class of distributions has a natural relationship with log-concave densities.
Key words and phrases: Log-concave, maximum likelihood estimation, multivariate data, probability mass function estimation, shape-constrained methods.