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Maximum likelihood estimation for totally positive log-concave densities. (English) Zbl 1473.62048

Summary: We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP\(_2\)) distributions and log-\(L^{\natural}\)-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 \(n \geq 3\). 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 \(\mathbb{R}^2\) under \(MTP_2\), and for samples in \(\mathbb{Q}^d\) under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.

MSC:

62E10 Characterization and structure theory of statistical distributions
62G07 Density estimation

Software:

LogConcDEAD
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