Modelling the distribution of the cluster maxima of exceedances of subasymptotic thresholds

A standard approach to model the extreme values of a stationary process is the peaks over threshold method, which consists of imposing a high threshold, identifying clusters of exceedances of this threshold and fitting the maximum value from each cluster using the generalized Pareto distribution. This approach is strongly justified by underlying asymptotic theory. We propose an alternative model for the distribution of the cluster maxima that accounts for the subasymptotic theory of extremes of a stationary process. This new distribution is a product of two terms, one for the marginal distribution of exceedances and the other for the dependence structure of the exceedance values within a cluster. We illustrate the improvement in fit, measured by the root mean square error of the estimated quantités, offered by the new distribution over the peaks over thresholds analysis using simulated and hydrological data, and we suggest a diagnostic tool to help identify when the proposed model is likely to lead to an improved fit.