scholarly journals Swimming performance index based on extreme value theory

2018 ◽  
Vol 14 (1) ◽  
pp. 51-62 ◽  
Author(s):  
Daniel T Gomes ◽  
Lígia Henriques-Rodrigues
Keyword(s):  
Extreme Value Theory ◽  
Performance Index ◽  
Pareto Distribution ◽  
Swimming Performance ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Selection Procedures ◽  
Generalized Pareto ◽  
Distribution Index

The International Swimming Federation has developed a points system that allows comparisons of results between different events. Such system is important for several reasons, since it is used as a criterion to rank swimmers in awards and selection procedures of national teams. The points system is based entirely on the world record of the correspondent event. Since it is based on only one observation, this work aims to suggest a new system, based on the probability distribution of the best performances in each event. Using extreme value theory, such distribution, under certain conditions, converges to a generalized Pareto distribution. The new performance index, based on the peaks over threshold methodology, is obtained based on the exceedance probabilities correspondent to the swimmers’ times that exceed a given threshold. We work with 17 officially recognized events in 50 m pool, for each women and men, and considered all-time rankings for all events until 31 December 2016. A study on the adequacy of the proposed generalized Pareto distribution index and a comparison between the performances of Usain Bolt and Michael Phelps are also conducted.

scholarly journals Using Statistical Approaches to Model Natural Disasters

2016 ◽  
Vol 13 (2) ◽  
Author(s):  
Audrene Edwards ◽  
Kumer Das
Keyword(s):  
Extreme Events ◽  
Extreme Value Theory ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Threshold Values ◽  
Data Set ◽  
Generalized Pareto ◽  
Estimated Parameters

The study of extremes has attracted the attention of scientists, engineers, actuaries, policy makers, and statisticians for many years. Extreme value theory (EVT) deals with the extreme deviations from the median of probability distributions and is used to study rare but extreme events. EVT’s main results characterize the distribution of the sample maximum or the distribution of values above a given threshold. In this study, EVT has been used to construct a model on the extreme and rare earthquakes that have happened in the United States from 1700 to 2011.The primary goal of fitting such a model is to estimate the amount of losses due to those extreme events and the probabilities of such events. Several diagnostic methods (for example, QQ plot and Mean Excess Plot) have been used to justify that the data set follows generalized Pareto distribution (GPD). Three estimation techniques have been employed to estimate parameters. The consistency and reliability of estimated parameters have been observed for different threshold values. The purpose of this study is manifold: first, we investigate whether the data set follows GPD, by using graphical interpretation and hypothesis testing. Second, we estimate GPD parameters using three different estimation techniques. Third, we compare consistency and reliability of estimated parameters for different threshold values. Last, we investigate the bias of estimated parameters using a simulation study. The result is particularly useful because it can be used in many applications (for example, disaster management, engineering design, insurance industry, hydrology, ocean engineering, and traffic management) with a minimal set of assumptions about the true underlying distribution of a data set. KEYWORDS: Extreme Value Theory; QQ Plot; Mean Excess Plot; Mean Residual Plot; Peak Over Threshold; Generalized Pareto Distribution; Maximum Likelihood Method; Method of Moments; Probability-Weighted Moments; Shapiro-Wilk test; Anderson- Darling Test

scholarly journals Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory

1997 ◽  
Vol 27 (1) ◽  
pp. 117-137 ◽  
Author(s):  
Alexander J. McNeil
Keyword(s):  
Extreme Value Theory ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Parametric Curve ◽  
Excess Loss ◽  
Generalized Pareto ◽  
Fire Insurance ◽  
Loss Severity ◽  
Parametric Curve Fitting

AbstractGood estimates for the tails of loss severity distributions are essential for pricing or positioning high-excess loss layers in reinsurance. We describe parametric curve-fitting methods for modelling extreme historical losses. These methods revolve around the generalized Pareto distribution and are supported by extreme value theory. We summarize relevant theoretical results and provide an extensive example of their application to Danish data on large fire insurance losses.

Using Extreme Value Theory to Model Electricity Price Risk with an Application to the Alberta Power Market

2005 ◽  
Vol 23 (5) ◽  
pp. 375-403 ◽  
Author(s):  
W. D. Walls ◽  
Wei. Zhang
Keyword(s):  
Extreme Value Theory ◽  
Value At Risk ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Financial Assets ◽  
Historical Simulation ◽  
Generalized Pareto ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed

Value-at-risk (VaR) is a measure of the maximum potential change in value of a portfolio of financial assets with a given probability over a given time horizon. VaR has become a standard measure of market risk and a common practice is to compute VaR by assuming that changes in value of the portfolio are conditionally normally distributed. However, assets returns usually come from heavy-tailed distributions, so computing VaR under the assumption of conditional normality can be an important source of error. We illustrate in our application to competitive electric power prices in Alberta, Canada, that VaR estimates based on extreme value theory models, in particular the generalized Pareto distribution are, more accurate than those produced by alternative models such as normality or historical simulation.

Stochastic comparison of parallel systems with Pareto components

2021 ◽  
pp. 1-13
Author(s):  
Sameen Naqvi ◽  
Weiyong Ding ◽  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Extreme Value Theory ◽  
Pareto Distribution ◽  
Parallel Systems ◽  
Value Theory ◽  
Extreme Value ◽  
Stochastic Comparison ◽  
Shape Parameters ◽  
Scale Parameters ◽  
Dispersive Ordering

Abstract Pareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$ , $a > 0$ . As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

A generalized Pareto distribution–based extreme value model of thermal gradients in a long-span bridge combining parameter updating

2016 ◽  
Vol 20 (2) ◽  
pp. 202-213 ◽  
Author(s):  
Guang-Dong Zhou ◽  
Ting-Hua Yi ◽  
Bin Chen ◽  
Huan Zhang
Keyword(s):  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Extreme Value ◽  
Thermal Gradients ◽  
Long Span Bridges ◽  
Long Span ◽  
Generalized Pareto ◽  
Long Span Bridge ◽  
Extreme Value Model ◽  
Value Model

Estimating extreme value models with high reliability for thermal gradients is a significant task that must be completed before reasonable thermal loads and possible thermal stress in long-span bridges are evaluated. In this article, a generalized Pareto distribution–based extreme value model combining parameter updating has been developed to describe the statistical characteristics of thermal gradients in a long-span bridge. The procedure of excluding correlation and the approach of selecting a proper threshold are suggested to prepare samples for generalized Pareto distribution estimation. A Bayesian estimation, which has the capability of updating model parameters by fusing prior information and incoming monitoring data, is proposed to fit the generalized Pareto distribution–based model. Furthermore, the Gibbs sampling, which is a Markov chain Monte Carlo algorithm, is adopted to derive the Bayesian posterior distribution. Finally, the proposed method is applied to the field monitoring data of thermal gradients in the Jiubao Bridge. The extreme value models of thermal gradients for the Jiubao Bridge are established, and the extreme thermal gradients with different return periods are extrapolated. The results indicate that the generalized Pareto distribution–based extreme value model has a strong ability to represent the statistical features of thermal gradients for the Jiubao Bridge, and the Bayesian estimation combining parameter updating provides high-precision generalized Pareto distribution–based models for predicting extreme thermal gradients. The predicted extreme thermal gradients are expected to evaluate and design long-span bridges.

Rate of convergence for the generalized Pareto approximation of the excesses

2003 ◽  
Vol 35 (04) ◽  
pp. 1007-1027 ◽  
Author(s):  
J.-P. Raoult ◽  
R. Worms
Keyword(s):  
Distribution Function ◽  
Uniform Convergence ◽  
Rate Of Convergence ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Generalized Pareto

Let F be a distribution function in the domain of attraction of an extreme-value distribution H γ. If F u is the distribution function of the excesses over u and G γ the distribution function of the generalized Pareto distribution, then it is well known that F u (x) converges to G γ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅ u (x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

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