Statistics for modeling heavy tailed distributions in geology: Part I. Methodology

Many natural phenomena exhibit size distributions that are power laws or power law type distributions Power laws are specific in the sense that they can exhibit extremely long or heavy tails. The largest event in a sample from such distribution usually dominates the underlying physical or generating process (floods, earthquakes, diamond sizes and values, incomes, insurance). Often, the practitioner is faced with the difficult problem of predicting values far beyond the highest sample value and designing his "system" either to profit from them, or to protect against extreme quantiles. In this paper we present a novel approach to estimating such heavy tails. The estimation of tail characteristics such as the extreme value index, extreme quantiles, and percentiles (rare events) is shown to depend primarily on the number of extreme data that are used to model the tail. Because only the most extreme data are useful for studying tails, thresholds must be selected above which the data are modeled as power laws. The mean square error (MSE) is used to select such thresholds. A semiparametric bootstrap method is developed to study estimation bias and variance and to derive confidence limits. A simulation study is performed to assess the accuracy of these confidence limits. The overall methodology is applied to the Harvard Central Moment Tensor catalog of global earthquakes.