Maximum Likelihood Estimation of Discrete Log-Concave Distributions with Applications

Shape-constrained methods specify a class of distributions instead of a single parametric family. The approach increases the robustness of the estimation without much loss of efficiency. Among these, log-concavity is an appealing shape constraint in distribution modeling, because it falls into the popular unimodal shape-constraint and many parametric models are log-concave. This is, therefore, the focus of our work.

First, we propose a maximum likelihood estimator of discrete log-concave distributions in higher dimensions. We define a new class of log-concave distributions in multiple dimensional spaces and study its properties. We show how to compute the maximum likelihood estimator 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, the proposed estimator outperforms a purely nonparametric approach (the empirical distribution), but is able to remain comparable to the correct parametric approach. Furthermore, the new class has a natural relationship with log-concave densities when data has been grouped or discretized. We show how this property can be used in a real data example.

Secondly, we apply the discrete log-concave maximum likelihood estimator in one-dimensional space to a clustering problem. Our work mainly focuses on the categorical nominal data. We develop a log-concave mixture model using the discrete log-concave maximum likelihood estimator. We then apply the log-concave mixture model to our clustering algorithm. We compare our proposed clustering algorithm with the other two clustering methods. Comparing results show that our proposed algorithm has a good performance.