Infinitely imbalanced binomial regression and deformed exponential families

• Behavior of binomial regression models for highly imbalanced data is studied.
• It is shown that the limit becomes a deformed exponential family.
• The proof uses a well-known result in the extreme value theory.
• A penalized maximum likelihood estimator is proposed with this.

The logistic regression model is known to converge to a Poisson point process model if the binary response tends to infinity 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.