Characterization of the Frequency of Extreme Earthquake Events by the Generalized Pareto Distribution

Recent results in extreme value theory suggest a new technique for statistical estimation of distribution tails (Embrechts et al., 1997), based on a limit theorem known as the Gnedenko-Pickands-Balkema-de Haan theorem. This theorem gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD), which is a family of distributions with two parameters. The GPD has been successfully applied in a number of statistical problems related to finance, insurance, hydrology, and other domains. Here, we apply the GPD approach to the well-known seismological problem of earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law. We analyze shallow earthquakes (depth h<70 km) in the Harvard catalog over the period 1977–2000 in 12 seismic zones. The GPD is found to approximate the tails of the seismic moment distributions quite well over the lower threshold approximately M 10²⁴ dyne-cm, or somewhat above (i.e., moment-magnitudes larger than m _W =5.3). We confirm that the b-value is very different (b=2.06 ± 0.30) in mid-ocean ridges compared to other zones (b=1.00 ± 0.04) with a very high statistical confidence and propose a physical mechanism contrasting crack-type rupture with dislocation-type behavior. The GPD can as well be applied in many problems of seismic hazard assessment on a regional scale. However, in certain cases, deviations from the GPD at the very end of the tail may occur, in particular for large samples signaling a novel regime.     

An extreme value analysis of the flow of Burbage Brook

An extreme value analysis of the flow of Burbage Brook is carried out by modelling peaks over a high threshold. The aims are to illustrate recently developed statistical techniques and to report on interesting features of the flow of the brook over a 58-year period. Peak flows are found to show marked seasonal variation and a downward trend. Then-year return level is estimated for various values ofn, and the reliability of the estimates is assessed.     

Linear combinations of order statistics to estimate the position and scale parameters of the Generalized Pareto distribution

 Ad hoc techniques for estimating the position and the scale parameters of the Generalized Pareto distribution are introduced. The estimators proposed are simple linear combinations of the order statistics: they provide valuable estimates of the parameters of interest, both when the shape parameter is known and when it is unknown (this latter case being of great relevance in practical applications), and show a good performance as well when the sample size is small. The procedures are tested on simulated data, and comparisons with other techniques are shown.     

An extreme value analysis of the flow of Burbage Brook

An extreme value analysis of the flow of Burbage Brook is carried out by modelling peaks over a high threshold. The aims are to illustrate recently developed statistical techniques and to report on interesting features of the flow of the brook over a 58-year period. Peak flows are found to show marked seasonal variation and a downward trend. Then-year return level is estimated for various values ofn, and the reliability of the estimates is assessed.     

Linear combinations of order statistics to estimate the quantiles of generalized pareto and extreme values distributions

 Ad hoc techniques for estimating the quantiles of the Generalized Pareto (GP) and the Generalized Extreme Values (GEV) distributions are introduced. The estimators proposed are based on new estimators of the position and the scale parameters recently introduced in the Literature. They provide valuable estimates of the quantiles of interest both when the shape parameter is known and when it is unknown (this latter case being of great relevance in practical applications). In addition, weakly-consistent estimators are introduced, whose calculation does not require the knowledge of any parameter. The procedures are tested on simulated data, and comparisons with other techniques are shown. The research was partially supported by Contract n. ENV4-CT97-0529 within the project “FRAMEWORK” of the European Community – D.G. XII. Grants by “Progetto Giovani Ricercatori” are also acknowledged.     

A note on the statistics of the largest geomagnetic storms per solar cycle

This note points out a problem with the way in which extreme value distributions have been fit to the intensities of the largest geomagnetic storms per solar cycle. An alternative method is described. This method is applied to observations of the three largest geomagnetic storms in solar cycles 11–22.     

A length-biased version of the Birnbaum-Saunders distribution with application in water quality

In this article, we develop a new model based on the Birnbaum-Saunders distribution that results to be both useful and practical for environmental sciences. The density, distribution and hazard functions, moments and properties of this new model are presented. A graphical analysis of the density is also provided. Furthermore, we estimate parameters, propose asymptotic inference and discuss influence diagnostics by using likelihood methods for the new distribution. An illustrative example with real data related to water quality indicates the adequacy on the new distribution.     

Diatoms from the genus Achnanthidium in flowing waters of the Appalachian Mountains (North America): Ecology, distribution and taxonomic notes

Diatoms from the genus Achnanthidium are abundant in rivers, streams, and springs of the Appalachian Mountains. They inhabit clean and polluted waters, including those affected by acid mine drainage. The identification of Achnanthidium taxa is difficult due to their small cell size and insufficient information in the diatom floras. We studied the taxonomy and ecology of Achnanthidium in Appalachian rivers by analyzing a data set of benthic diatom samples and corresponding water chemistry data collected during several water-quality surveys from 181 sampling sites. Ten species were identified using scanning electron and light microscopy: A. alpestre (Lowe & Kociolek) Lowe & Kociolek, A. atomus (Hustedt) Monnier, Lange-Bertalot, & Ector, A. deflexum (Reimer) Kingston, A. duthii (Sreenivasa) Edlund, A. eutrophilum (Lange-Bertalot) Lange-Bertalot, A. cf. gracillimum (Meister) Lange-Bertalot, A. cf. latecephalum Kobayasi, A. minutissimum (Kützing) Czarnecki (sensu lato), A. reimeri (Camburn) comb. nov., and A. rivulare Potapova & Ponader. The distribution of common taxa in relation to water chemistry was studied by fitting non-parametric regression models (generalized additive models, GAM, and non-parametric multiplicative regression models, NPMR) to species relative abundances. Studied Achnanthidium species differed considerably in their responses to water chemistry. These results suggest that species-level identifications will lead to more accurate bioassessments.