scholarly journals Optimal heavy tail estimation – Part 1: Order selection

2017 ◽  
Vol 24 (4) ◽  
pp. 737-744 ◽  
Author(s):  
Manfred Mudelsee ◽  
Miguel A. Bermejo
Keyword(s):  
Time Series ◽  
Heavy Tails ◽  
Characteristic Exponent ◽  
Tail Probability ◽  
Heavy Tail ◽  
Infinite Variance ◽  
Accurate Estimation ◽  
Monte Carlo Experiment ◽  
River Elbe ◽  
Tail Estimation

Abstract. The tail probability, P, of the distribution of a variable is important for risk analysis of extremes. Many variables in complex geophysical systems show heavy tails, where P decreases with the value, x, of a variable as a power law with a characteristic exponent, α. Accurate estimation of α on the basis of data is currently hindered by the problem of the selection of the order, that is, the number of largest x values to utilize for the estimation. This paper presents a new, widely applicable, data-adaptive order selector, which is based on computer simulations and brute force search. It is the first in a set of papers on optimal heavy tail estimation. The new selector outperforms competitors in a Monte Carlo experiment, where simulated data are generated from stable distributions and AR(1) serial dependence. We calculate error bars for the estimated α by means of simulations. We illustrate the method on an artificial time series. We apply it to an observed, hydrological time series from the River Elbe and find an estimated characteristic exponent of 1.48 ± 0.13. This result indicates finite mean but infinite variance of the statistical distribution of river runoff.

scholarly journals Optimal heavy tail estimation, Part I: Order selection

2017 ◽  
Author(s):  
Manfred Mudelsee ◽  
Miguel A. Bermejo
Keyword(s):  
Time Series ◽  
Heavy Tails ◽  
Characteristic Exponent ◽  
Tail Probability ◽  
Heavy Tail ◽  
Infinite Variance ◽  
Accurate Estimation ◽  
Monte Carlo Experiment ◽  
River Elbe ◽  
Tail Estimation

Abstract. The tail probability, P, of the distribution of a variable is important for risk analysis of extremes. Many variables in complex geophysical systems show heavy tails, where P decreases with the value, x, of a variable as a power law with characteristic exponent, α. Accurate estimation of α on the basis of data is currently hindered by the problem of the selection of the order, that is, the number of largest x-values to utilize for the estimation. This paper presents a new, widely applicable, data-adaptive order selector, which is based on computer simulations and brute force search. It is the first in a set of papers on optimal heavy tail estimation. The new selector outperforms competitors in a Monte Carlo experiment, where simulated data are generated from stable distributions and AR(1) serial dependence. We calculate error bars for the estimated α by means of simulations. We illustrate the method on an artificial time series. We apply it to an observed, hydrological time series from the river Elbe and find an estimated characteristic exponent of 1.48 ± 0.13. This result indicates finite mean but infinite variance of the statistical distribution of river runoff.

ESTIMATION OF AND INFERENCE ABOUT THE EXPECTED SHORTFALL FOR TIME SERIES WITH INFINITE VARIANCE

2013 ◽  
Vol 29 (4) ◽  
pp. 771-807 ◽  
Author(s):  
Oliver Linton ◽  
Zhijie Xiao
Keyword(s):  
Time Series ◽  
Emerging Market ◽  
Expected Shortfall ◽  
Infinite Variance ◽  
Monte Carlo Experiment ◽  
Dependence Structure ◽  
Market Exchange ◽  
Finite Sample ◽  
Inference Procedure ◽  
Thickness Parameter

We study estimation and inference of the expected shortfall for time series with infinite variance. Both the smoothed and nonsmoothed estimators are investigated. The rate of convergence is determined by the tail thickness parameter, and the limiting distribution is in the stable class with parameters depending on the tail thickness parameter of the time series and on the dependence structure, which makes inference complicated. A subsampling procedure is proposed to carry out statistical inference. We also analyze a nonparametric estimator of the conditional expected shortfall. A Monte Carlo experiment is conducted to evaluate the finite sample performance of the proposed inference procedure, and an empirical application to emerging market exchange rates (from October 1997 to October 2008) is conducted to highlight the proposed study.

DYNAMICAL ESTIMATION OF CALENDAR EFFECTS

2006 ◽  
Vol 06 (01) ◽  
pp. L7-L15
Author(s):  
ALEXANDROS LEONTITSIS
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Noise Level ◽  
Nonlinear Dynamical System ◽  
Main Tool ◽  
Experimental Results ◽  
Accurate Estimation ◽  
Nonlinear Dynamical ◽  
Correlation Integral ◽  
Calendar Effects

The paper introduces a method for estimation and reduction of calendar effects from time series, which their fluctuations are governed by a nonlinear dynamical system and additive normal noise. Calendar effects can be considered deviations of the distribution(s) of particular group(s) of observations that have a common characteristic related to the calendar. The concept of this method is the following: since the calendar effects are not related to the dynamics of the time series, the accurate estimation and reduction will result a time series with a smaller amount of noise level (i.e. more accurate dynamics). The main tool of this method is the correlation integral, due to its inherit capability of modeling both the dynamics and the additive normal noise. Experimental results are presented on the Nasdaq Cmp. index.

scholarly journals Preliminaries on the Accurate Estimation of the Hurst Exponent Using Time Series

2021 ◽  
Author(s):  
Ginno Millán ◽  
Román Osorio-Comparán ◽  
Gastón Lefranc
Keyword(s):  
Time Series ◽  
Hurst Exponent ◽  
High Speed ◽  
Mean Squared Error ◽  
Computer Network ◽  
Accurate Estimation ◽  
Minimum Length ◽  
Squared Error ◽  
In Series ◽  
Speed Computer

<div>This article explores the required amount of time series points from a high-speed computer network to accurately estimate the Hurst exponent. The methodology consists in designing an experiment using estimators that are applied to time series addresses resulting from the capture of high-speed network traffic, followed by addressing the minimum amount of point required to obtain in accurate estimates of the Hurst exponent. The methodology addresses the exhaustive analysis of the Hurst exponent considering bias behaviour, standard deviation, and Mean Squared Error using fractional Gaussian noise signals with stationary increases. Our results show that the Whittle estimator successfully estimates the Hurst exponent in series with few</div><div>points. Based on the results obtained, a minimum length for the time series is empirically proposed. Finally, to validate the results, the methodology is applied to real traffic captures in a high-speed computer network.</div>

scholarly journals Inhomogeneous phase-type distributions and heavy tails

2019 ◽  
Vol 56 (4) ◽  
pp. 1044-1064 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Mogens Bladt
Keyword(s):  
Real World ◽  
Heavy Tails ◽  
Heavy Tail ◽  
Transition Rates ◽  
Modelling Framework ◽  
Inhomogeneous Phase ◽  
Fire Insurance ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Heavy Tailed

AbstractWe extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable the carrying over of fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach in modelling a real-world heavy-tailed fire insurance dataset.

scholarly journals Scaling of High-Quantile Estimators

2011 ◽  
Vol 48 (04) ◽  
pp. 968-983 ◽  
Author(s):  
Matthias Degen ◽  
Paul Embrechts
Keyword(s):  
Heavy Tails ◽  
Global Financial Crisis ◽  
Financial Institution ◽  
Small Sample ◽  
Accurate Estimation ◽  
Risk Capital ◽  
High Quantile ◽  
Quantile Estimators ◽  
Small Sample Sizes ◽  
The Global Financial Crisis

Enhanced by the global financial crisis, the discussion about an accurate estimation of regulatory (risk) capital a financial institution needs to hold in order to safeguard against unexpected losses has become highly relevant again. The presence of heavy tails in combination with small sample sizes turns estimation at such extreme quantile levels into an inherently difficult statistical issue. We discuss some of the problems and pitfalls that may arise. In particular, based on the framework of second-order extended regular variation, we compare different high-quantile estimators and propose methods for the improvement of standard methods by focusing on the concept of penultimate approximations.

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