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.

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 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

Modeling European hot spells using extreme value analysis

2014 ◽  
Vol 58 (3) ◽  
pp. 193-207 ◽  
Author(s):  
C Photiadou ◽  
MR Jones ◽  
D Keellings ◽  
CF Dewes
Keyword(s):  
Extreme Value ◽  
Extreme Value Analysis ◽  
Value Analysis ◽  
Hot Spells

scholarly journals Threshold selection in univariate extreme value analysis

Extremes ◽  
2021 ◽  
Author(s):  
Laura Fee Schneider ◽  
Andrea Krajina ◽  
Tatyana Krivobokova
Keyword(s):  
Mean Squared Error ◽  
Extreme Value ◽  
Hill Estimator ◽  
Small Samples ◽  
Extreme Value Analysis ◽  
Threshold Selection ◽  
Value Analysis ◽  
Wide Range ◽  
Asymptotic Mean Squared Error ◽  
The Hill

AbstractThreshold selection plays a key role in various aspects of statistical inference of rare events. In this work, two new threshold selection methods are introduced. The first approach measures the fit of the exponential approximation above a threshold and achieves good performance in small samples. The second method smoothly estimates the asymptotic mean squared error of the Hill estimator and performs consistently well over a wide range of processes. Both methods are analyzed theoretically, compared to existing procedures in an extensive simulation study and applied to a dataset of financial losses, where the underlying extreme value index is assumed to vary over time.

Spatial stochastic simulation to aid local extreme value analysis of cyclone-induced wave heights when numerical hydrodynamic simulations are scarce

2021 ◽  
Author(s):  
Jeremy Rohmer ◽  
Rodrigo Pedreros ◽  
Yann Krien
Keyword(s):  
Confidence Interval ◽  
Stochastic Simulation ◽  
Extreme Value ◽  
Return Level ◽  
Extreme Value Analysis ◽  
Percentage Error ◽  
Value Analysis ◽  
Return Levels ◽  
Wave Heights ◽  
Local Extreme

<p>To estimate return levels of wave heights (Hs) induced by tropical cyclones at the coast, a commonly-used approach is to (1) randomly generate a large number of synthetic cyclone events (typically >1,000); (2) numerically simulate the corresponding Hs over the whole domain of interest; (3) extract the Hs values at the desired location at the coast and (4) perform the local extreme value analysis (EVA) to derive the corresponding return level. Step 2 is however very constraining because it often involves a numerical hydrodynamic simulator that can be prohibitive to run: this might limit the number of results to perform the local EVA (typically to several hundreds). In this communication, we propose a spatial stochastic simulation procedure to increase the database size of numerical results with synthetic maps of Hs that are stochastically generated. To do so, we propose to rely on a data-driven dimensionality-reduction method, either unsupervised (Principal Component Analysis) or supervised (Partial Least Squares Regression), that is trained with a limited number of pre-existing numerically simulated Hs maps. The procedure is applied to the Guadeloupe island and results are compared to the commonly-used approach applied to a large database of Hs values computed for nearly 2,000 synthetic cyclones (representative of 3,200 years – Krien et al., NHESS, 2015). When using only a hundred of cyclones, we show that the estimates of the 100-year return levels can be achieved with a mean absolute percentage error (derived from a bootstrap-based procedure) ranging between 5 and 15% around the coasts while keeping the width of the 95% confidence interval of the same order of magnitude than the one using the full database. Without synthetic Hs maps augmentation, the error and confidence interval width are both increased by nearly 100%. A careful attention is paid to the tuning of the approach by testing the sensitivity to the spatial domain size, the information loss due to data compression, and the number of cyclones. This study has been carried within the Carib-Coast INTERREG project (https://www.interreg-caraibes.fr/carib-coast).</p>

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