The L Distribution and Skew Generalizations

Abstract: Leptokurtic or platykurtic distributions can, for example, be gen-erated by applying certain non-linear transformations to a Gaussian random variable. Within this work we focus on the class of so-called power trans-formations which are determined by their generator function. Examples are the H−transformation of Tukey (1960), the J−transformation of Fischer and Klein (2004) and the L−transformation which is derived from Johnson’s in-verse hyperbolic sine transformation. It is shown that generator functions themselves which meet certain requirements can be used to construct both probability densities and cumulative distribution functions. For the J−trans-formation, we recover the logistic distribution. Using the L−transformation, a new class of densities is derived, discussed and generalized