Maximum Likelihood Estimation for Totally Positive Log-Concave Densities
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We study nonparametric density estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely multivariate totally positive distributions of order 2 (MTP_2, a.k.a. log-supermodular) and the subclass of log-L^-concave (LLC) distributions. In both cases we impose the additional assumption of log-concavity in order to ensure boundedness of the likelihood function. Given n independent and identically distributed random vectors from a d-dimensional MTP_2 distribution (LLC distribution, respectively), we show that the maximum likelihood estimator (MLE) exists and is unique with probability one when n≥ 3 (n≥ 2, respectively), independent of the number d of variables. The logarithm of the MLE is a tent function in the binary setting and in R^2 under MTP_2 and in the rational setting under LLC. We provide a conditional gradient algorithm for computing it, and we conjecture that the same convex program also yields the MLE in the remaining cases.
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