Kurtosis modelling by means of the J-transformation

Summary: The H–family of distributions or H–distributions, introduced by Tukey (1960; 1977), aregenerated by a single transformation of the standard normal distribution and allow for leptokurtosisrepresented by the parameter h. Alternatively, Haynes et al. (1997) generated leptokurtic distributionsby applying the K–transformation to the normal distribution. In this study we propose a third transformation,the so–called J–transformation, and derive some properties of this transformation. Moreover,so-called elongation generating functions (EGFs) are introduced. By means of EGFs we are able tovisualize the strength of tail elongation and to construct new transformations. Finally, we compare thethree transformations towards their goodness–of–fit in the context of financial return data.