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Volume 24, issue 4 | Copyright
Nonlin. Processes Geophys., 24, 737-744, 2017
https://doi.org/10.5194/npg-24-737-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 06 Dec 2017

Research article | 06 Dec 2017

Optimal heavy tail estimation – Part 1: Order selection

Manfred Mudelsee1,2 and Miguel A. Bermejo1 Manfred Mudelsee and Miguel A. Bermejo
  • 1Climate Risk Analysis, Heckenbeck, Bad Gandersheim, Germany
  • 2Alfred Wegener Institute Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany

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.

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Risk analysis of extremes has high socioeconomic relevance. Of crucial interest is the tail probability, P, of the distribution of a variable, which is the chance of observing a value equal to or greater than a certain threshold value, x. Many variables in geophysical systems (e.g. climate) show heavy tail behaviour, where P may be rather large. In particular, P decreases with x as a power law that is described by a parameter, α. We present an improved method to estimate α on data.
Risk analysis of extremes has high socioeconomic relevance. Of crucial interest is the tail...
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