Power kurtosis transformations: Definition, properties and ordering

摘要

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 the H-transformation of Tukey (1960), the K-transformation of MacGillivray and Cannon (1997) and the J-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.