Alfarano, Simone and Lux, Thomas (2010): Extreme Value Theory as a Theoretical Background for Power Law Behavior.

PDF
MPRA_paper_24718.pdf Download (274kB)  Preview 
Abstract
Power law behavior has been recognized to be a pervasive feature of many phenomena in natural and social sciences. While immense research efforts have been devoted to the analysis of behavioral mechanisms responsible for the ubiquity of powerlaw scaling, the strong theoretical foundation of power laws as a very general type of limiting behavior of large realizations of stochastic processes is less well known. In this chapter, we briefly present some of the key results of extreme value theory, which provide a statistical justification for the emergence of power laws as limiting behavior for extreme fluctuations. The remarkable generality of the theory allows to abstract from the details of the system under investigation, and therefore allows its application in many diverse fields. Moreover, this theory offers new powerful techniques for the estimation of the Pareto index, detailed in the second part of this chapter.
Item Type:  MPRA Paper 

Original Title:  Extreme Value Theory as a Theoretical Background for Power Law Behavior 
Language:  English 
Keywords:  Extreme Value Theory; Power Laws; Tail index 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C13  Estimation: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C14  Semiparametric and Nonparametric Methods: General C  Mathematical and Quantitative Methods > C0  General > C01  Econometrics 
Item ID:  24718 
Depositing User:  Simone Alfarano 
Date Deposited:  30. Aug 2010 19:30 
Last Modified:  30. Dec 2015 08:57 
References:  1. J. Beirlant, P. Vynckier, and J.L. Teugels. Practical Analysis of Extreme Values. University Press, Leuven, 1996. 2. JP Bouchaud and M. Potters. Theory of Financial Risks. From Statistical Physics to Risk Managment. Univerity press, Cambrige, 2000. 3. R.B. D’Agostino and M.A. Stephens. GoodnessofFit Techniques. MarcelDekker, inc., New York, 1986. 4. J. Danielsson, L. de Hann, Peng L., and C.G. de Vries. Using a bootstrap method to choose the otimal sample fraction in tail index estimation. Journal of Multivariate Analysis, 76:226–248, 2001. 5. J. Danielsson and C.G. de Vries. Tail index and quantile estimation with very high frequency data. Journal of Empirical Finance, 4:241–257, 1997. 6. H. Drees and E. Kaufmann. Selecting the optimal sample fraction in univariate extreme value estimation. Stochastic Processes and Their Applications, 75:149–172, 1998. 7. W. Feller. An Introduction to Probability Theory and Its Applications. John Wiley and Sons, New York, 1971. 8. C.M. Goldie and R.L. Smith. Slow variation with remainder: Theory and applications. Quarterly Journal of Mathematics, 38:45–71, 1987. 9. P. Hall. On some simple estimates of an exponent of regular variation. Journal of the Royal Statistical Society, Series B, 44:37–42, 1982. 10. P. Hall. Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems. Journal of Multivariate Analysis, 32:177–203, 1990. 11. B. M. Hill. A simple general approach to inference about the tail of a distribution. Annals of Statistics, 3:1163–1173, 1975. 12. P. Kearns and A. Pagan. Estimating the density tail index for financial time series. Review of Economics and Statistics, 79:171–175, 1997. 13. T. Lux. The limiting extremal behavior of speculative returns: an analysis of intradaily data from the Frankfurt stock exchange. Applied Financial Economics, 11:299–315, 1996. 14. T. Lux. On moment condition failure in german stock returns: an application of recent advances in extreme value statistics. Empirical Economics, 20:641–652, 2000. 15. D.M. Mason. Laws of large numbers for sums of extreme values. Annals of Probability, 10:754–764, 1982. 16. R.D. Reiss and M. Thomas. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Birkh¨auser, Basle, 1997. 17. H. Rootzen, M.R. Leadbetter, and L. de Hann. On the distribution of tail array sums for strongly mixing stationary sequences. Annals of Applied Probability, 8:868–885, 1998. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/24718 