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 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 Extreme events in total ozone over Arosa – Part 1: Application of extreme value theory

2010 ◽  
Vol 10 (20) ◽  
pp. 10021-10031 ◽  
Author(s):  
H. E. Rieder ◽  
J. Staehelin ◽  
J. A. Maeder ◽  
T. Peter ◽  
M. Ribatet ◽  
...  
Keyword(s):  
Time Series ◽  
Frequency Distribution ◽  
Extreme Events ◽  
Extreme Value Theory ◽  
Total Ozone ◽  
Volcanic Eruptions ◽  
Value Theory ◽  
Extreme Value ◽  
Data Set ◽  
Ozone Data

Abstract. In this study ideas from extreme value theory are for the first time applied in the field of stratospheric ozone research, because statistical analysis showed that previously used concepts assuming a Gaussian distribution (e.g. fixed deviations from mean values) of total ozone data do not adequately address the structure of the extremes. We show that statistical extreme value methods are appropriate to identify ozone extremes and to describe the tails of the Arosa (Switzerland) total ozone time series. In order to accommodate the seasonal cycle in total ozone, a daily moving threshold was determined and used, with tools from extreme value theory, to analyse the frequency of days with extreme low (termed ELOs) and high (termed EHOs) total ozone at Arosa. The analysis shows that the Generalized Pareto Distribution (GPD) provides an appropriate model for the frequency distribution of total ozone above or below a mathematically well-defined threshold, thus providing a statistical description of ELOs and EHOs. The results show an increase in ELOs and a decrease in EHOs during the last decades. The fitted model represents the tails of the total ozone data set with high accuracy over the entire range (including absolute monthly minima and maxima), and enables a precise computation of the frequency distribution of ozone mini-holes (using constant thresholds). Analyzing the tails instead of a small fraction of days below constant thresholds provides deeper insight into the time series properties. Fingerprints of dynamical (e.g. ENSO, NAO) and chemical features (e.g. strong polar vortex ozone loss), and major volcanic eruptions, can be identified in the observed frequency of extreme events throughout the time series. Overall the new approach to analysis of extremes provides more information on time series properties and variability than previous approaches that use only monthly averages and/or mini-holes and mini-highs.

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.

scholarly journals Extreme events in total ozone over Arosa – Part 1: Application of extreme value theory

2010 ◽  
Vol 10 (5) ◽  
pp. 12765-12794 ◽  
Author(s):  
H. E. Rieder ◽  
J. Staehelin ◽  
J. A. Maeder ◽  
T. Peter ◽  
M. Ribatet ◽  
...  
Keyword(s):  
Time Series ◽  
Frequency Distribution ◽  
Extreme Events ◽  
Extreme Value Theory ◽  
Total Ozone ◽  
Volcanic Eruptions ◽  
Value Theory ◽  
Extreme Value ◽  
Data Set ◽  
Ozone Data

Abstract. In this study ideas from extreme value theory are for the first time applied in the field of stratospheric ozone research, because statistical analysis showed that previously used concepts assuming a Gaussian distribution (e.g. fixed deviations from mean values) of total ozone data do not adequately address the structure of the extremes. We show that statistical extreme value methods are appropriate to identify ozone extremes and to describe the tails of the Arosa (Switzerland) total ozone time series. In order to accommodate the seasonal cycle in total ozone, a daily moving threshold was determined and used, with tools from extreme value theory, to analyse the frequency of days with extreme low (termed ELOs) and high (termed EHOs) total ozone at Arosa. The analysis shows that the Generalized Pareto Distribution (GPD) provides an appropriate model for the frequency distribution of total ozone above or below a mathematically well-defined threshold, thus providing a statistical description of ELOs and EHOs. The results show an increase in ELOs and a decrease in EHOs during the last decades. The fitted model represents the tails of the total ozone data set with high accuracy over the entire range (including absolute monthly minima and maxima), and enables a precise computation of the frequency distribution of ozone mini-holes (using constant thresholds). Analyzing the tails instead of a small fraction of days below constant thresholds provides deeper insight into the time series properties. Fingerprints of dynamical (e.g. ENSO, NAO) and chemical features (e.g. strong polar vortex ozone loss), and major volcanic eruptions, can be identified in the observed frequency of extreme events throughout the time series. Overall the new approach to analysis of extremes provides more information on time series properties and variability than previous approaches that use only monthly averages and/or mini-holes and mini-highs.

Predicting Federal Funds Rate Using Extreme Value Theory

2020 ◽  
Vol 35 (1) ◽  
pp. 1-15
Author(s):  
Ashim Kumar Dey ◽  
Kumer Pial Das
Keyword(s):  
Extreme Events ◽  
Extreme Value Theory ◽  
Simulated Data ◽  
Value Theory ◽  
Extreme Value ◽  
Economic Sectors ◽  
Federal Funds ◽  
Federal Funds Rate ◽  
Estimated Parameters ◽  
The Us

AbstractThe extreme value theory (EVT) is used to assess the risk of extreme events caused by natural calamities or untoward circumstances in the social and economic sectors. The theory can be used to study the frequency of rare events and to build up a predictive model so that one can attempt to forecast the frequency of such future extreme events such as a financial collapse and the amount of damage from such a collapse. Even though many statistical techniques have been used to analyze the manner in which the Federal Reserve determines the level of the Federal Fund Rates, no known study has used EVT to analyze and predict the extreme fund rates. In this study, the US Federal Funds Rate, one of the most publicized and important economic indicators in the financial world, from 1954–2019 has been analyzed. The contributions of this study are: (1) to provide an appropriate model for the normalized Federal Funds Rate data; (2) to compare several estimation techniques in estimating parameters for two possible models; (3) to predict the maximum economic return rate from a Federal Funds Rate in the future by using the concept of the return period; and (4) to investigate the bias of estimated parameters applying a simulation study. Simulated data and real financial data are used for the study, and the outcome satisfies the efficiency of its application.

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.

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