Plot of the average cost of exceedances against threshold, whose slope immediately reveals whether a tail is exponential, light, or heavy, and from which threshold it stabilizes.
For each threshold, compute the mean excess of the losses above it and plot that mean against the threshold. The resulting chart reads in a second. Flat, and the tail is exponential, because the exponential law is memoryless and the average cost of what exceeds does not depend on the threshold. Rising in a straight line with positive slope, and the tail is Pareto, with the slope giving the tail index directly, which is an independent check on the Hill estimator. Falling, and the tail is lighter than exponential and bounded. It is the best threshold selection tool for the peaks over threshold method, since the threshold to keep is the one from which the plot turns linear and stays linear. Reading it takes one precaution: the last points rest on two or three observations and swing wildly, so you do not interpret them.
Property losses in an industrial book, 2012 to 2025. The mean excess plot is flat around 180K EUR up to an 800K EUR threshold, then rises linearly with a slope of 0.45 beyond. Reading: the body of the distribution is near exponential, the tail turns Pareto above 800K EUR, and the tail index implied by the slope is consistent with the Hill estimator measured separately.
mean excess plot, fonction d'excès moyen, mean residual life plot, courbe des excédents moyens