Modelling of discrete extremes through extended versions of discrete generalized Pareto distribution

The statistical modelling of integer-valued extremes such as large avalanche counts has received less attention than their continuous counterparts in the extreme value theory (EVT) literature. One approach to moving from continuous to discrete extremes is to model threshold exceedances of integer random variables by the discrete version of the generalized Pareto distribution. Still, the optimal threshold selection that defines exceedances remains a problematic issue. Moreover, within a regression framework, the treatment of the many data points (those below the chosen threshold) is either ignored or decoupled from extremes. Considering these issues, we extend the idea of using a smooth transition between the two tails (lower and upper) to force large and small discrete extreme values to comply with EVT. In the case of zero inflation, we also develop models with an additional parameter. To incorporate covariates, we extend the Generalized Additive Models (GAM) framework to discrete extreme responses. In the GAM forms, the parameters of our proposed models are quantified as a function of covariates. The maximum likelihood estimation procedure is implemented for estimation purposes. With the advantage of bypassing the threshold selection step, our findings indicate that the proposed models are more flexible and robust than competing models (i.e. discrete generalized Pareto distribution and Poisson distribution).