Extremes, Extreme Spacings and Tail Lengths : An Investigation for Some Important Distributions

Abstract

Extreme value and extreme spacing distributions are elegant and important artifacts of statistical theory and practice. However, in statistical education, due to the highly technical nature of the subject, they are generally treated as special topics. But, as demonstrated by Freimer et al. (1989), the asymptotic distributions of the extremes and extreme spacings of random samples, and the related theory can be derived and developed by applying elementary methods to the population quantile functions when they are available in closed forms. However, their work excluded the pedagogically important Gaussian and gamma populations. In this paper, using the closed form expression for the quantile function of the Pareto family first, we show how this approach works. We then proceed to demonstrate its use by simple Taylor expansion for the normal and gamma populations, cases where the closed form expressions for the quantile functions are unavailable. In the process, we relate the geometric notion of tail length to the extreme value distribution. We also examine the case of inverse Gaussian (IG) family, which is well known to be strikingly and intriguingly analogous to the Gaussian family. Actually, in the present context we consider not only the IG but the related reciprocal IG (RIG) and the root reciprocal IG (RRIG) families. The extreme value theory for these three families is derived, again using the elementary methods, even though their quantile functions also lack closed form expressions. Interestingly, it is seen that the extreme value theory for the RRIG population, and not of the IG population, is somewhat analogous to the Gaussian distribution. AMS (2000) Subject Classification : 62E20.