Infinitely imbalanced binomial regression and deformed exponential families

TL;DR

This paper explores the universal behavior of binomial regression models under extreme class imbalance, revealing convergence to Poisson point processes and the emergence of deformed exponential families for various link functions.

Contribution

It demonstrates the universality of the Poisson point process limit across a broad class of link functions in binomial regression and introduces a penalized maximum likelihood estimator.

Findings

Logit, probit, and cloglog links lead to exponential family intensities.

Other link functions result in deformed exponential families.

Theoretical proof relies on extreme value theory.

Abstract

The logistic regression model is known to converge to a Poisson point process model if the binary response tends to infinitely imbalanced. In this paper, it is shown that this phenomenon is universal in a wide class of link functions on binomial regression. The proof relies on the extreme value theory. For the logit, probit and complementary log-log link functions, the intensity measure of the point process becomes an exponential family. For some other link functions, deformed exponential families appear. A penalized maximum likelihood estimator for the Poisson point process model is suggested.