Method that keeps only losses above a high threshold and fits a generalized Pareto distribution to their excesses, more efficient than block maxima because it discards none of them.
Two ways of isolating extremes coexist. Block maxima keeps only the largest loss of each year, which throws away a year with three catastrophes and keeps a quiet one. Peaks over threshold instead keeps everything above a threshold, hence all the available extreme information, and models the excesses with a generalized Pareto distribution. The problem then becomes threshold choice, and it is a bias against variance tradeoff that no formula settles. Too low, and you pull in losses from the body of the distribution where the asymptotic result does not apply, so the shape parameter is biased. Too high, and fifteen observations remain and the estimator turns unstable. Two graphical tools guide the choice: the mean excess plot, which should turn linear above the right threshold, and a plot of the estimated shape parameter against threshold, where you look for the first stable plateau.
Motor bodily injury book, losses from 2005 to 2025. At a 300K EUR threshold, 612 exceedances and a shape parameter of 0.19; at 1M EUR, 143 exceedances and 0.34; at 2M EUR, 61 exceedances and 0.36; at 4M EUR, 22 exceedances and 0.33 with a very wide interval. The plateau starts at 1M EUR, which is the threshold selected, and the gap between 0.19 and 0.35 doubles the estimated 99.5th percentile.
POT, méthode POT, peaks-over-threshold, excès au-dessus d'un seuil