On the Asymptotic Properties of Hogg's Q Statistic

Abstract

The asymptotic properties of Q_n, a measure of tail thickness introduced by Hogg, are investigated. The nondegenerate limit distribution of Q_n is obtained as an application of a bivariate extension of a result on smooth linear functions of order statistics, due to Stigler. The limiting result is helpful in indicating the optimum proportion of the sample data to be used in the construction of Q_n for discriminating between members of a symmetric family of distributions. The asymptotic theory is also used to find a close approximation to the expected value of Q_n. This approximation works well for moderate sample sizes for distributions in the symmetric family.