Generalized Tukey-type distributions with application to financial and teletraffic data

Constructing skew and heavy-tailed distributions by transforming a standard normal variable goes back to Tukey (Exploratory data analysis. Addison-Wesley, Reading, 1977) and was extended and formalized by Hoaglin (In: Data analysis for tables, trends, and shapes. Wiley, New York, 1983) and Martinez and Iglewicz (Commun Statist Theory Methods 13(3):353–369, 1984). Applications of Tukey’s GH distribution family—which are composed by a skewness transformation G and a kurtosis transformation H—can be found, for instance, in financial, environmental or medical statistics. Recently, alternative transformations emerged in the literature. Rayner and MacGillivray (Statist Comput 12:57–75, 2002b) discuss the GK distributions, where Tukey’s H-transformation is replaced by another kurtosis transformation K. Similarly, Fischer and Klein (All Stat Arch, 88(1):35–50, 2004) advocate the J-transformation which also produces heavy tails but—in contrast to Tukey’s H-transformation—still guarantees the existence of all moments. Within this work we present a very general kurtosis transformation which nests H-, K-and an approximation to the J-transformation and, hence, permits to discriminate between them. Applications to financial and teletraffic data are given.