Abstract
An extreme value approach to the modeling of rare and damaging events quite frequently involves heavy tailed distributions associated with power decaying tails. The positive counterpart of this power, which determines the tail heaviness of the distribution function pertaining to the sample observations, is consensually known as the tail index. In this paper, we allow the tail index $α $ to be zero so as to embrace the class of super-heavy tailed distributions. We then present a test statistic consisting of the ratio of maximum to the sum of log-excesses in order to discern between distributions with heavy and super-heavy tails. Under suitable yet reasonable assumptions, we cast an account of consistency of the Hill estimator for α equal to zero from the asymptotic features of the referred testing procedure.
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References
Beirlant, J. and Teugels, J. L. T. (1989). Asymptotic normality of Hill’s estimator. InExtreme Value Theory Proceedings (Oberwolfach, 1987, Lecture Notes in Statistics), Vol.51, Springer, New York.
Embrechts, P., Klüuppelberg, C. and Mikosch, T. (1997).Modelling Extremal Events for Insurance and Finance. Springer, New York.
Fisher, R. A. and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest and smallest member of a sample.Cambridge Philosophical Society. Mathematical Proceedings, 24:180–190.
Fraga Alves, M. I. de Haan, L. and Neves, C. (2006). Statistical inference for heavy and super-heavy tailed distributions.Available at: http://people.few.eur.nl/ ldehaan/.
Geluk, J. L. and de Haan, L. (1987).Regular Variation, Extensions and Tauberian theorems. CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, The Netherlands.
Geluk, J. L. and de Haan, L. (2000). Stable probability distributions and their domains of attraction: a direct approach.Probability and Mathematical Statistics, 20:169–188.
de Haan, L. (1970).On regular variation and its application to the weak convergence of sample extremes. Mathematisch Centrum Amsterdam.
de Haan, L. and Ferreira, A. (2006).Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering.
Toppozada, T., Branum, D., Petersen, M., Hallstrom, C., Cramer, C., and Reichle, M. (2000). Epicenters of and areas damaged by M > 5 California earthquakes, 1800-1999 (CDMG Map Sheet 49); Updated (3/2004) with data from: Toppozada, T. R. and D. Branum (2002): California M > 5.5 earthquakes, history and areas damaged, in Lee, W. H., Kanamori, H. and Jennings, P.International Handbook of Earthquake and Engineering Seismology, International Association of Seismology and Physics of the Earth’s Interior; National Earthquake Information Center (http://neic.usgs.gov/).
Zaliapin, I. V., Kagan, Y. Y. and Schoenberg, F. P. (2005). Approximating the Distribution of Pareto Sums.Pure and Applied Geophysics, 162:1187–1228.
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© 2008 Birkhäuser Boston
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Neves, C., Alves, I.F. (2008). Ratio of Maximum to the Sum for Testing Super Heavy Tails. In: Advances in Mathematical and Statistical Modeling. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4626-4_13
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DOI: https://doi.org/10.1007/978-0-8176-4626-4_13
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Online ISBN: 978-0-8176-4626-4
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