scholarly journals Extreme Value and Record Statistics in Heavy-Tailed Processes With Long-Range Memory

2012 ◽  
pp. 315-334 ◽  
Author(s):  
Aicko Y. Schumann ◽  
Nicholas R. Moloney ◽  
Jörn Davidsen
Keyword(s):  
Long Range ◽  
Extreme Value ◽  
Record Statistics ◽  
Heavy Tailed

scholarly journals Extreme Value Analysis for Mixture Models with Heavy-Tailed Impurity

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2208
Author(s):  
Ekaterina Morozova ◽  
Vladimir Panov
Keyword(s):  
Stock Returns ◽  
Mixture Models ◽  
Extreme Value ◽  
Extreme Value Analysis ◽  
Value Analysis ◽  
Numerical Examples ◽  
Proposed Model ◽  
Mixing Parameter ◽  
Heavy Tailed ◽  
Regularly Varying Distribution

This paper deals with the extreme value analysis for the triangular arrays which appear when some parameters of the mixture model vary as the number of observations grows. When the mixing parameter is small, it is natural to associate one of the components with “an impurity” (in the case of regularly varying distribution, “heavy-tailed impurity”), which “pollutes” another component. We show that the set of possible limit distributions is much more diverse than in the classical Fisher–Tippett–Gnedenko theorem, and provide the numerical examples showing the efficiency of the proposed model for studying the maximal values of the stock returns.

TAIL AND NONTAIL MEMORY WITH APPLICATIONS TO EXTREME VALUE AND ROBUST STATISTICS

2011 ◽  
Vol 27 (4) ◽  
pp. 844-884 ◽  
Author(s):  
Jonathan B. Hill
Keyword(s):  
Least Squares ◽  
Central Limit ◽  
Robust Statistics ◽  
Tail Dependence ◽  
Extreme Value ◽  
Limit Theory ◽  
Trimmed Sums ◽  
Garch Processes ◽  
Heavy Tailed ◽  
Tail Events

New notions of tail and nontail dependence are used to characterize separately extremal and nonextremal information, including tail log-exceedances and events, and tail-trimmed levels. We prove that near epoch dependence (McLeish, 1975; Gallant and White, 1988) and L0-approximability (Pötscher and Prucha, 1991) are equivalent for tail events and tail-trimmed levels, ensuring a Gaussian central limit theory for important extreme value and robust statistics under general conditions. We apply the theory to characterize the extremal and nonextremal memory properties of possibly very heavy-tailed GARCH processes and distributed lags. This in turn is used to verify Gaussian limits for tail index, tail dependence, and tail-trimmed sums of these data, allowing for Gaussian asymptotics for a new tail-trimmed least squares estimator for heavy-tailed processes.

scholarly journals EXTREME VALUE ANALYSIS WITHOUT THE LARGEST VALUES: WHAT CAN BE DONE?

2019 ◽  
Vol 34 (2) ◽  
pp. 200-220
Author(s):  
Jingjing Zou ◽  
Richard A. Davis ◽  
Gennady Samorodnitsky
Keyword(s):  
Order Statistics ◽  
Extreme Values ◽  
Sample Path ◽  
Estimation Procedure ◽  
Extreme Value ◽  
Hill Estimator ◽  
Extreme Value Analysis ◽  
Value Analysis ◽  
Heavy Tailed ◽  
The Hill

AbstractIn this paper, we are concerned with the analysis of heavy-tailed data when a portion of the extreme values is unavailable. This research was motivated by an analysis of the degree distributions in a large social network. The degree distributions of such networks tend to have power law behavior in the tails. We focus on the Hill estimator, which plays a starring role in heavy-tailed modeling. The Hill estimator for these data exhibited a smooth and increasing “sample path” as a function of the number of upper order statistics used in constructing the estimator. This behavior became more apparent as we artificially removed more of the upper order statistics. Building on this observation we introduce a new version of the Hill estimator. It is a function of the number of the upper order statistics used in the estimation, but also depends on the number of unavailable extreme values. We establish functional convergence of the normalized Hill estimator to a Gaussian process. An estimation procedure is developed based on the limit theory to estimate the number of missing extremes and extreme value parameters including the tail index and the bias of Hill's estimator. We illustrate how this approach works in both simulations and real data examples.

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.

scholarly journals Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series

Extremes ◽  
2016 ◽  
Vol 19 (3) ◽  
pp. 517-547 ◽  
Author(s):  
Richard A. Davis ◽  
Johannes Heiny ◽  
Thomas Mikosch ◽  
Xiaolei Xie
Keyword(s):  
Time Series ◽  
Multivariate Time Series ◽  
Extreme Value ◽  
Extreme Value Analysis ◽  
Value Analysis ◽  
Heavy Tailed ◽  
Autocovariance Matrices ◽  
Sample Autocovariance

scholarly journals Trimmed extreme value estimators for censored heavy-tailed data

2021 ◽  
Vol 15 (1) ◽  
Author(s):  
Martin Bladt ◽  
Hansjörg Albrecher ◽  
Jan Beirlant
Keyword(s):  
Extreme Value ◽  
Heavy Tailed
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