Maximum Likelihood Estimation for Totally Positive Log‐Concave Densities
Name
1806.10120.pdf
Description
Accepted version
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921.04 KB
Format
Adobe PDF
Checksum (MD5)
163dac38e0d31b49e6f492dd4bac20a6
Author(s)
Robeva, Elina
Uhler, Caroline
Date Issued
April 2020
Journal
Scandinavian Journal of Statistics
Publisher
Wiley
Citation
Robeva, Elina et al. “Maximum Likelihood Estimation for Totally Positive Log‐Concave Densities.” Scandinavian Journal of Statistics (April 2020) © 2020 The Author(s)
Version
Author's final manuscript
Abstract
We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP2) 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 in (Formula presented.) under MTP2, and for samples in (Formula presented.) under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Institute for Data, Systems, and Society
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
10.1111/SJOS.12462
