Measures of local variation in a distribution: Expected length of spacings and variances of order statistics

Abstract

Two measures of variation about the r th order statistic X r:n are studied: the expected value of the spacing X r+1:n −X r:n and the variance of X r:n . The nature of the dependence of these measures on r , for n fixed, is examined for various classes of distributions. In the case of the variance of X r:n the situations considered are supplemented by some asymptotic results.