Tukey - type distributions in the context of financial data

Using the Gaussian distribution as statistical model for data sets is widely spread, especially in practice. However, departure from normality seems to be more the rule than the exception. The H-distributions, introduced by Tukey (1960, 1977), are generated by a single transformation (H-transformation) of a standard normal distribution (or, more general, of a symmetric distribution) Z and allow for leptokurtosis represented by the (elongation) parameter h > 0. In order to additionally take skewness into account by means of certain transformations, several generalizations and extensions (HQ,HH, GH,GK, . . .) have been proposed in the literature. Within this work we ”complete” this class of Tukey-type distributions by introducing KQand JQ-distributions on the one side and KK−, JJ− and G̃J−distributions on the other side. Moreover, we empirically compare the goodness-of-fit of such Tukey-type distributions for different symmetrical distributions Z (here: Gaussian, logistic and hyperbolic secant distribution) in the context of financial return data. In particular, the interplay between Z and the Tukey-type transformations is investigated. Finally, results are compared to those of popular multi-parametric distribution models with closedform densities.