scholarly journals Estimating -Functionals for Heavy-Tailed Distributions and Application

2010 ◽  
Vol 2010 ◽  
pp. 1-34 ◽  
Author(s):  
Abdelhakim Necir ◽  
Djamel Meraghni
Keyword(s):  
Asymptotic Normality ◽  
Extreme Value Theory ◽  
Financial Risk ◽  
Risk Measures ◽  
Value Theory ◽  
Statistical Parameters ◽  
Linear Combinations ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Actuarial Risk

-functionals summarize numerous statistical parameters and actuarial risk measures. Their sample estimators are linear combinations of order statistics (-statistics). There exists a class of heavy-tailed distributions for which the asymptotic normality of these estimators cannot be obtained by classical results. In this paper we propose, by means of extreme value theory, alternative estimators for -functionals and establish their asymptotic normality. Our results may be applied to estimate the trimmed -moments and financial risk measures for heavy-tailed distributions.

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.

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.

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.

scholarly journals Estimating the Conditional Tail Expectation in the Case of Heavy-Tailed Losses

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Abdelhakim Necir ◽  
Abdelaziz Rassoul ◽  
Ričardas Zitikis
Keyword(s):  
Risk Assessment ◽  
Asymptotic Normality ◽  
Financial Risk ◽  
Risk Measure ◽  
Extreme Value ◽  
Second Moment ◽  
Heavy Tailed ◽  
Actuarial Risk ◽  
Conditional Tail Expectation

The conditional tail expectation (CTE) is an important actuarial risk measure and a useful tool in financial risk assessment. Under the classical assumption that the second moment of the loss variable is finite, the asymptotic normality of the nonparametric CTE estimator has already been established in the literature. The noted result, however, is not applicable when the loss variable follows any distribution with infinite second moment, which is a frequent situation in practice. With a help of extreme-value methodology, in this paper, we offer a solution to the problem by suggesting a new CTE estimator, which is applicable when losses have finite means but infinite variances.

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.

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