An extended generalized Pareto regression model for count data

The statistical modelling of discrete extremes has received less attention than their continuous counterparts in the extreme value theory (EVT) literature. One approach to the transition from continuous to discrete extremes is the modelling of threshold exceedances of integer random variables by the discrete version of the generalized Pareto distribution. However, the optimal choice of thresholds defining exceedances remains a problematic issue. Moreover, in a regression framework, the treatment of the majority of non-extreme data below the selected threshold is either ignored or separated from the extremes. To tackle these issues, we expand on the concept of employing a smooth transition between the bulk and the upper tail of the distribution. In the case of zero inflation, we also develop models with an additional parameter. To incorporate possible predictors, we relate the parameters to additive smoothed predictors via an appropriate link, as in the generalized additive model (GAM) framework. A penalized maximum likelihood estimation (MLE) procedure is implemented. We illustrate our modelling proposal with a real dataset of avalanche activity in the French Alps. With the advantage of bypassing the threshold selection step, our results indicate that the proposed models are more flexible and robust than competing models, such as the negative binomial distribution.