  This paper studies binary logistic regression for rare events data, or
imbalanced data, where the number of events (observations in one class, often
called cases) is significantly smaller than the number of nonevents
(observations in the other class, often called controls). We first derive the
asymptotic distribution of the maximum likelihood estimator (MLE) of the
unknown parameter, which shows that the asymptotic variance convergences to
zero in a rate of the inverse of the number of the events instead of the
inverse of the full data sample size. This indicates that the available
information in rare events data is at the scale of the number of events instead
of the full data sample size. Furthermore, we prove that under-sampling a small
proportion of the nonevents, the resulting under-sampled estimator may have
identical asymptotic distribution to the full data MLE. This demonstrates the
advantage of under-sampling nonevents for rare events data, because this
procedure may significantly reduce the computation and/or data collection
costs. Another common practice in analyzing rare events data is to over-sample
(replicate) the events, which has a higher computational cost. We show that
this procedure may even result in efficiency loss in terms of parameter
estimation.
