Maximum likelihood estimation for totally positive log‐concave densities
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 under MTP2, and for samples in under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Sep 14, 2020
- Source ID
- 10.1111/sjos.12462
Entities
People
- Bernd Sturmfels
- Caroline Uhler
- Elina Robeva
- Ngoc Tran
Organizations
- Alfred P. Sloan Foundation
- Massachusetts Institute of Technology
- National Science Foundation Division of Mathematical Sciences
- Office of Naval Research Global
- University of British Columbia
- University of California
- University of Texas at Austin