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.
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 plus first, and its reciprocal estimates the tail index of the underlying Pareto law. It requires no parametric fit and takes one line to compute, which makes it the first move in any tail analysis. Its weakness is the same as threshold choice, and here it is visible to the eye: plot the estimator against k and the resulting 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 does not mean the estimator has failed, it means the tail is not a power law, and that information is as useful 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 equal to 20, forms a plateau between k equal to 35 and k equal to 70 around 1.7, then falls toward 1.1. The retained 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.
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