Some Properties of the Range in Samples from Tukey's Symmetric Lambda Distributions

Abstract

Tukey introduced a family of random variables defined by the transformation Z = [U^λ - (1 - U)^γ]/λ where U is uniformly distributed on [0, 1]. Some of its properties are described with emphasis on properties of the sample range. The rectangular and logistic distributions are members of this family and distributions corresponding to certain values of λ give useful approximations to the normal and t distributions. Closed form expressions are given for the expectation and coefficient of variation of the range and numerical values are computed for n = 2(1)6(2)12, 15, 20 for several values of λ. It is observed that Plackett's upper bound on the expectation of the range for samples of size n is attained for a λ distribution with λ = n − 1.