  Our contribution is to widen the scope of extreme value analysis applied to
discrete-valued data. Extreme values of a random variable $X$ are commonly
modeled using the generalized Pareto distribution, a method that often gives
good results in practice. When $X$ is discrete, we propose two other methods
using a discrete generalized Pareto and a generalized Zipf distribution
respectively. Both are theoretically motivated and we show that they perform
well in estimating rare events in several simulated and real data cases such as
word frequency, tornado outbreaks and multiple births.
