A note on the construction of generalized Tukey-type transformations

One possibility to construct heavy tail distributions is to directly manipulate a
standard Gaussian random variable by means of transformations which satisfy
certain conditions. This approach dates back to Tukey (1960) who introduces the
popular H-transformation. Alternatively, the K-transformation of MacGillivray
& Cannon (1997) or the J-transformation of Fischer & Klein (2004) may be used.
Recently, Klein & Fischer (2006) proposed a very general power kurtosis transformation
which includes the above-mentioned transformations as special cases.
Unfortunately, their transformation requires an in nite number of unknown parameters
to be estimated. In contrast, we introduce a very simple method to
construct exible kurtosis transformations. In particular, manageable “superstructures”
are suggested in order to statistically discriminate between H-, J– and
K-distributions (associated to H-, J– and K-transformations).