Extremes,extreme spacings and outliers in the tukey and weibull families

Abstract

In this paper we analyze the tail character of the generalized Tukey lambda and extended Weibull distributions and in the process demonstrate that the details of convergence in law of the extreme spacings are graphically descriptive of the tail-lengths and the associated outlier proneness. In terms of the classical extreme value theory, it is seen that the two shape parameters of the generalized Tukey lambda family separately determine the nature of the two tails individually and that the family is sparse in medium tailed distributions. For the extended Weibull (or generalized extreme value) family on the other hand, the right tail is always medium, while the left tail can be either long or short. An analysis based upon the limiting distribution of extreme spacings especially O _p ·, the high probability order of magnitude for large samples, is used to refine and clarify the classical tail classification. The derivations involve elementary enough methods to make the results pedagogically suitable èven at introductory level.

Keywords:

• extreme value distributions
• order statistics
• quantile functions
• tail behavior