Extreme value analysis of diamond-size distributions

Extreme value analysis provides a semiparametric method for analyzing the extreme long tails of skew distributions which may be observed when handling mining data. The estimation of important tail characteristics, such as the extreme value index, allows for a discrimination between competing distribution models. It measures the thickness of such long tailed distributions, if only a limited sample is available. This paper stresses the practical implementation of extreme value theory, which is used to discriminate a lognormal from a mixed lognormal distribution in a case study of size distributions for alluvial diamonds.     

Statistics for Modeling Heavy Tailed Distributions in Geology: Part II. Applications

In this paper, we present three diverse types of applications of extreme value statistics in geology, namely: earthquakes magnitudes, diamond values, and impact crater size distribution on terrestrial planets. Each of these applications has a different perspective toward tail modeling, yet many of these phenomena exhibit heavy or long tails which can be modeled by power laws. It is shown that the estimation of important tail characteristics, such as the extreme value index, is directly linked to the interpretation of the underlying geological process. Only the most extreme data are useful for studying such phenomena, so thresholds must be selected above which the data become power laws. In the case of earthquake magnitudes, we investigate the use of extreme value statistics in predicting large events on the global scale and for shallow intracontinental earthquakes in Asia. Large differences are found between estimates obtained from extreme value statistics and the usually applied standard statistical techniques. In the case of diamond deposits, we investigate the impact of the most precious stones in the global valuation of primary deposits. It is shown that in the case of Pareto-type behavior, the expected value of few extreme stones in the entire deposit has considerable influence on the global valuation. In the case of impact crater distributions, we study the difference between craters distributions on Earth and Mars and distributions occurring on other planets or satellites within the solar system. A striking result is that all planets display the same distributional tail except for Earth and Mars. In a concluding account, we demonstrate the apparent loghyperbolic variation in all of the above-mentioned examples.     

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.     

Estimating the maximum possible earthquake magnitude using extreme value methodology: the Groningen case

Natural Hazards - Both the river network and the regions outside the estuary mouths in the Pearl River Delta (PRD) of China experienced significant changes from 1999 to 2014. A validated...