Mean excess plot
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.
Davison and Smith, 1990, Journal of the Royal Statistical Society, Series B: the peaks over threshold method fits a generalized Pareto distribution to the excesses above a threshold, and everything hinges on the threshold chosen. The mean excess plot is how you choose it. For each threshold, compute the mean excess of the losses above it and plot that mean against the threshold. The plot reads at a glance. Flat, and the tail is exponential: the exponential law is memoryless, so the average cost of what exceeds does not depend on the threshold. Rising in a straight line, and the tail is Pareto, with the slope giving the tail index directly, an independent check on the Hill estimator. Falling, and the tail is lighter than exponential and bounded. Keep the threshold from which the plot turns linear and stays linear. One precaution: the last points rest on two or three observations and swing wildly, so they are ignored.
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. Interpretation: the body of the distribution is close to exponential, the tail turns Pareto above 800K EUR, and the tail index implied by the slope is consistent with the Hill estimator measured separately.
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