Peaks over threshold

Two ways of isolating extremes coexist. Block maxima keeps only the largest loss of each year, so if three catastrophes hit the same year, two of them vanish, while the largest loss of a quiet year is kept. 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, a bias-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 hold, and the shape parameter is biased. Too high, and only fifteen observations remain, so the estimator turns unstable. Two graphical tools guide the choice. The mean excess plot should turn linear above the right threshold; on a plot of the estimated shape parameter against threshold, you look for the first stable plateau.