Summary
Heavy tail distributions can be generated by applying specific non-linear transformations to a Gaussian random variable. Within this work we introduce power kurtosis transformations which are essentially determined by their generator function. Examples are theH-transformation of Tukey (1960), theK-transformation of MacGillivray and Cannon (1997) and theJ-transformation of Fischer and Klein (2004).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 van Zwet (1964) for symmetric distributions.
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References
Balanda, K.P., MacGillivray, H.L. (1990). Kurtosis and spread.Canadian Journal of Statistics 18 17–30.
Fama, E. (1965). The behaviour of stock prices.Journal of Business 38 34–105.
Fischer, M., Klein, I. (2004). Kurtosis modelling by means of the j-transformation.Allgemeines Statistisches Archiv 88 1–16.
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.
MacGillivray, H. L., Cannon, W. (1997). Generalizations of the g-and-h distributions and their uses. Unpublished.
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.
Van Zwet, W.R. (1964). Convex transformations of random variables. Mathematical Centre Tracts No. 7, Mathematical Centre, Amsterdam.
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Klein, I., Fischer, M. Power kurtosis transformations: Definition, properties and ordering. Allgemeines Statistisches Arch 90, 395–401 (2006). https://doi.org/10.1007/s10182-006-0241-1
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DOI: https://doi.org/10.1007/s10182-006-0241-1