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Logarithmic Voronoi polytopes for discrete linear models

Yulia Alexandr

Vol. 15 (2024), No. 1, 1–13
DOI: 10.2140/astat.2024.15.1
Abstract

We study logarithmic Voronoi cells for linear statistical models and partial linear models. The logarithmic Voronoi cells at points on such model are polytopes. To any d-dimensional linear model inside the probability simplex Δn1, we can associate an n × d matrix B. For interior points, we describe the vertices of these polytopes in terms of cocircuits of B. We also show that these polytopes are combinatorially isomorphic to the dual of a vector configuration with Gale diagram B. This means that logarithmic Voronoi cells at all interior points on a linear model have the same combinatorial type. We also describe logarithmic Voronoi cells at points on the boundary of the simplex. Finally, we study logarithmic Voronoi cells of partial linear models, where the points on the boundary of the model are especially of interest.

Keywords
maximum likelihood estimation, linear models, polytopes, logarithmic Voronoi cells, Gale diagrams, boundary, partial linear models, combinatorial type, statistics, probability simplex
Mathematical Subject Classification
Primary: 51M20, 52A40, 62F10
Milestones
Received: 14 December 2022
Revised: 10 June 2023
Accepted: 3 July 2023
Published: 11 January 2024
Authors
Yulia Alexandr
Department of Mathematics
University of California
Berkeley, CA
United States