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Maximum likelihood estimation for totally positive log‐concave densities
Author(s) -
Robeva Elina,
Sturmfels Bernd,
Tran Ngoc,
Uhler Caroline
Publication year - 2021
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12462
Subject(s) - mathematics , maximum likelihood , likelihood function , statistics , restricted maximum likelihood , exponential family , independent and identically distributed random variables , estimator , combinatorics , likelihood ratio test , conditional probability distribution , dimension (graph theory) , likelihood principle , nonparametric statistics , maximum likelihood sequence estimation , quasi maximum likelihood , random variable
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 ♮ ‐ 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 inℝ 2under MTP 2 , and for samples inℚ dunder LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.
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