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Box–Cox transformations and heavy-tailed distributions

Part of: Real functions Nonparametric inference Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jef L. Teugels  and
Giovanni Vanroelen
Jef L. Teugels
Affiliation:
University Statistics Centre (UCS), Katholieke Universiteit Leuven, W. De Croylaan 54, 3001 Leuven, Belgium. Email address: [email protected]
Giovanni Vanroelen
Affiliation:
University Statistics Centre (UCS), Katholieke Universiteit Leuven, W. De Croylaan 54, 3001 Leuven, Belgium. Email address: [email protected]
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Abstract

It is a stylized fact that estimators in extreme-value theory suffer from serious bias. Moreover, graphical representations of extremal data often show erratic behaviour. In the statistical literature it is advised to use a Box–Cox transformation in order to make data more suitable for statistical analysis. We provide some of the theoretical background to see the effect of such transformations and to investigate under what circumstances they might be helpful.

Keywords

Box–Cox transformationsecond-order regular variationHill estimatorheavy-tailed distributions

MSC classification

Primary: 26A12: Rate of growth of functions, orders of infinity, slowly varying functions
Secondary: 62G32: Statistics of extreme values; tail inference 60G70: Extreme value theory; extremal processes
Type
Part 4. Heavy-tail analysis
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 213 - 227
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Beirlant, J., Dierckx, G., Goegebeur, Y. and Matthys, G. (1999). Tail index estimation and an exponential regression model. Extremes 2, 177–200.Google Scholar
[2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
[3] De Haan, L. (1970). On Regular Variation and Its Applications to the Weak Convergence of Sample Extremes (Math. Centre Tracts 32). Mathematisch Centrum, Amsterdam.Google Scholar
[4] De Haan, L. and Stadtmüller, U. (1996). Generalized regular variation of second order. J. Austral. Math. Soc. A 61, 381–395.CrossRefGoogle Scholar
[5] Fisher, R. A. and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Camb. Phil. Soc. 24, 180–190.CrossRefGoogle Scholar
[6] Fraga Alves, M. I. (2001). A location invariant Hill-type estimator. Extremes 4, 199–217.Google Scholar
[7] Geluk, J. L. and De Haan, L. (1987). Regular Variation, Extensions and Tauberian Theorems (CWI Tract 40). Stichting Mathematisch Centrum, Amsterdam.Google Scholar
[8] Gnedenko, B. V. (1943). Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453.Google Scholar
[9] Gumbel, E. J. (1958). Statistics of Extremes. Columbia University Press, New York.Google Scholar
[10] Heyde, C. C. (1969). On extremal factorisation and recurrent events. J. R. Statist. Soc. B 31, 72–79.Google Scholar
[11] Heyde, C. C. (1997). Quasi-likelihood and Its Application: a General Approach to Optimal Parameter Estimation. Springer, New York.CrossRefGoogle Scholar
[12] Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163–1174.Google Scholar
[13] Matthys, G. (2001). Estimating the extreme-value index and high quantiles: bias reduction and optimal threshold selection. Doctoral Thesis, Katholieke Universiteit Leuven.Google Scholar
[14] Segers, J. (2001). Extremes of a random sample: limit theorems and statistical applications. Doctoral Thesis, Katholieke Universiteit Leuven.Google Scholar
[15] Vanroelen, G. (2003). The effect of transformations on second-order regular variation. Doctoral Thesis, Katholieke Universiteit Leuven.Google Scholar