Hill estimator
Nonparametric estimator of the tail index of a power-law distribution, computed on the k largest observations, whose plot against k serves as a diagnostic as much as a result.
Bruce Hill, Annals of Statistics, 1975. When a tail decays as a power law, the Hill estimator measures its exponent directly. Its formula is the mean of the logarithms of the k largest observations minus the logarithm of the (k+1)th, and its reciprocal estimates the tail index of the underlying Pareto distribution. It needs no parametric fit and takes one line to compute, which makes it the first move in any tail analysis. Its weakness is threshold choice, and you can see it. Plot the estimator against k. The Hill plot starts in a very noisy zone for small k, crosses a plateau, then drifts down as k grows and the body of the distribution enters the calculation. The plateau gives the answer. A Hill plot with no plateau says the tail is not a power law, and that is worth as much as the value you were after.
Reported cyber losses in a large-account book, 2018 to 2025, 470 claims. The Hill plot is unreadable below k = 20, forms a plateau between k = 35 and k = 70 around 1.7, then falls toward 1.1. The selected index is 1.7, a tail where variance does not exist but the mean does: an unlimited treaty stays priceable, a standard-deviation-based model does not.
Hill estimator, Hill plot, tail index estimator
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