Squared Linear Models

TL;DR

This paper investigates squared linear models, analyzing their critical points, algebraic structure, tropical degenerations, and connections to determinantal point processes, revealing their geometric and statistical properties.

Contribution

It provides a detailed algebraic and geometric analysis of squared linear models, including their likelihood ideal, singular locus, and tropical degenerations, with new determinantal representations.

Findings

All critical points are real and positive.

Each region of the projective hyperplane contains exactly one critical point.

The likelihood correspondence admits a determinantal presentation.

Abstract

We study statistical models that are parametrized by squares of linear forms. All critical points of the likelihood function are real and positive. There is one critical point in each region of the projective hyperplane arrangement defined by the linear forms. We examine the ideal and singular locus of the model, and we give a determinantal presentation for its likelihood correspondence. We characterize tropical degenerations of the MLE, we describe the log-normal polytopes, and we explore connections to determinantal point processes.