  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.
