scholarly journals Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach

2019 ◽  
Vol 42 (2) ◽  
pp. 143-166 ◽  
Author(s):  
Renato Santos Silva ◽  
Fernando Ferraz Nascimento
Keyword(s):  
Monte Carlo ◽  
Order Statistics ◽  
Bayesian Approach ◽  
Extreme Value Theory ◽  
Likelihood Estimation ◽  
Value Theory ◽  
Extreme Value ◽  
Environmental Data ◽  
Heavy Tailed ◽  
The Bayesian Approach

Extreme Value Theory (EVT) is an important tool to predict efficient gains and losses. Its main areas of analyses are economic and environmental. Initially, for that form of event, it was developed the use of patterns of parametric distribution such as Normal and Gamma. However, economic and environmental data presents, in most cases, a heavy-tailed distribution, in contrast to those distributions. Thus, it was faced a great difficult to frame extreme events. Furthermore, it was almost impossible to use conventional models, making predictions about non-observed events, which exceed the maximum of observations. In some situations EVT is used to analyse only the maximum of some dataset, which provide few observations, and in those cases it is more effective to use the r largest-order statistics. This paper aims to propose Bayesian estimators' for parameters of the r largest-order statistics. During the research, it was used Monte Carlo simulation to analyze the data, and it was observed some properties of those estimators, such as mean, variance, bias and Root Mean Square Error (RMSE). The estimation of the parameters provided inference for its parameters and return levels. This paper also shows a procedure to the choice of the r-optimal to the r largest-order statistics, based on the Bayesian approach applying Markov chains Monte Carlo (MCMC). Simulation results reveal that the Bayesian approach has a similar performance to the Maximum Likelihood Estimation, and the applications were developed using the Bayesian approach and showed a gain in accurary compared with otherestimators.

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.

Bayesian approach and extreme value theory in economic analysis of forestry projects

2019 ◽  
Vol 105 ◽  
pp. 64-71 ◽  
Author(s):  
Thiago Taglialegna Salles ◽  
Denismar Alves Nogueira ◽  
Luiz Alberto Beijo ◽  
Liniker Fernandes da Silva
Keyword(s):  
Economic Analysis ◽  
Bayesian Approach ◽  
Extreme Value Theory ◽  
Value Theory ◽  
Extreme Value

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.

The analysis of design wind speed estimates specified in the National Building Code of Canada

2007 ◽  
Vol 34 (4) ◽  
pp. 513-524 ◽  
Author(s):  
M D Pandey ◽  
Y An
Keyword(s):  
Wind Speed ◽  
Order Statistics ◽  
Extreme Value Theory ◽  
Gumbel Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Building Code ◽  
Peaks Over Threshold ◽  
Wind Speed Data ◽  
Design Wind Speed

The design wind pressures specified in the 2005 National Building Code of Canada (NBCC) have been derived from the Gumbel distribution fitted to annual maximum wind speed data collected up to early 1990s. The statistical estimates of the annual maxima method are affected by a relatively large sampling variability, since the method considers a fairly small subset of available wind speed records. Advanced statistical methods have emerged in recent years with the purpose of reducing both sampling and model uncertainties associated with extreme quantile estimates. The two most notable methods are the peaks-over-threshold (POT) and annually r largest order statistics (r-LOS), which extend the data set by including additional maxima observed in wind speed time series. The objective of the paper is to explore the use of advanced extreme value theory for updating the design wind speed estimates specified in the Canadian building design code. The paper re-examines the NBCC method for design wind speed estimation and presents the analysis of the latest Canadian wind speed data by POT, r-LOS, and annual maxima methods. The paper concludes that r-LOS method is an effective alternative for the estimation of extreme design wind speed.Key words: wind speed, extreme value theory, order statistics, return period, maximum likelihood method, peaks-over-threshold method, generalized extreme value distribution, Gumbel distribution, generalized Pareto distribution.

Box–Cox transformations and heavy-tailed distributions

2004 ◽  
Vol 41 (A) ◽  
pp. 213-227 ◽  
Author(s):  
Jef L. Teugels ◽  
Giovanni Vanroelen
Keyword(s):  
Statistical Analysis ◽  
Extreme Value Theory ◽  
Value Theory ◽  
Extreme Value ◽  
Theoretical Background ◽  
Graphical Representations ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Box Cox Transformation

It is a stylized fact that estimators in extreme-value theory suffer from serious bias. Moreover, graphical representations of extremal data often show erratic behaviour. In the statistical literature it is advised to use a Box–Cox transformation in order to make data more suitable for statistical analysis. We provide some of the theoretical background to see the effect of such transformations and to investigate under what circumstances they might be helpful.

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