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Power kurtosis transformations: definition, properties and ordering. (English) Zbl 1118.62013

Summary: Heavy tail distributions can be generated by applying specific nonlinear transformations to a Gaussian random variable. Within this work we introduce power kurtosis transformations which are essentially determined by their generator function. Examples are the \(H\)-transformation of J. W. Tukey [The practical relationship between the common transformation of percentages of counts and of amounts. Tech. Rep. No. 36, Princeton Univ. Stat. Tech. Res. Group (1960)], the \(K\)-transformation of H. L. MacGillivray and W. Cannon [Generalizations of the \(g\)- and \(h\)-distributions and their uses. Unpublished manuscript] and the \(J\)-transformation of the authors [Allg. Stat. Arch. 88, No. 1, 35–50 (2004; Zbl 1123.62303)]. Furthermore, we derive a general condition on the generator function which guarantees that the corresponding transformation is actually tail-increasing. In this case the exponent of the power kurtosis transformation can be interpreted as a kurtosis parameter. We also prove that the transformed distributions can be ordered with respect to the partial ordering of W. R. van Zwet [Convex transformations of random variables. Math. Centre Tracts. 7. (1964; Zbl 0125.37102)] for symmetric distributions.

MSC:

62E10 Characterization and structure theory of statistical distributions
62G32 Statistics of extreme values; tail inference
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References:

[1] Balanda, K.P., MacGillivray, H.L. (1990). Kurtosis and spread.Canadian Journal of Statistics 18 17–30. · Zbl 0692.62010 · doi:10.2307/3315414
[2] Fama, E. (1965). The behaviour of stock prices.Journal of Business 38 34–105. · doi:10.1086/294743
[3] Fischer, M., Klein, I. (2004). Kurtosis modelling by means of the j-transformation.Allgemeines Statistisches Archiv 88 1–16. · Zbl 1123.62303 · doi:10.1007/s101820400158
[4] Fischer, M., Horn, A., Klein, I. (2006). Tukey-type distributions in the context of financial return data.Communications in Statistics (Theory and Methods) to be published 2006. · Zbl 1118.62015
[5] MacGillivray, H. L., Cannon, W. (1997). Generalizations of the g-and-h distributions and their uses. Unpublished.
[6] Tukey, J.W. (1960). The practical relationship between the common transformation of percentages of counts and of ammounts. Technical Report No. 36, Princeton University Statistical Techniques Research Group.
[7] Van Zwet, W.R. (1964). Convex transformations of random variables. Mathematical Centre Tracts No. 7, Mathematical Centre, Amsterdam. · Zbl 0125.37102
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