Meaning
Nonparametric method used to determine the tail index of a heavy tailed distribution based on a set of upper order statistics. The hill estimator provides a way to measure the thickness of the tail without assuming a specific distribution shape. It relies on the largest values in a sample to make its calculation.
Arrangement Order
Sorting of data from smallest to largest allows the user to focus on the top tier of observations. For the hill estimator to work, one must choose a specific number of high value points to include in the average. This choice influences the stability of the final estimate.
Logarithmic Ratio
Numerical differences between the largest values are averaged to produce the result. The hill estimator uses this average to find the exponent that governs the decay of the tail. A lower exponent indicates a more dangerous distribution with more frequent extremes.
Balancing Act
Trade off between the variance of the estimate and the systematic error is a constant challenge. If too few points are used, the hill estimator becomes volatile, but if too many are used, it picks up data that does not belong to the tail. Finding the right cut off is a requirement for accurate measurement.
