Linear Functions of Order Statistics

Abstract

The purpose of this paper is to investigate the asymptotic normality of linear combinations of order statistics; that is, to find conditions under which a statistic of the form Sn = ∑n i=1 cinXin has a limiting normal distribution as n becomes infinite, where the cin's are constants and X1n, X2n, ⋯, Xnn are the observations of a sample of size n, ordered by increasing magnitude. Aside from the sample mean (the case where the weights cin are all equal to 1/n), the first proof of asymptotic normality within this class was by Smirnov in 1935 [19], who considered the case that nonzero weight is attached to at most two percentiles. In 1946, Mosteller [13] extended this to the case of several percentiles, and coined the phrase "systematic statistic" to describe Sn. Since the publication in 1955 of a paper by Jung [11] concerned with finding optimal weights for Sn in certain estimation problems, interest in proving its asymptotic normality under more general conditions has grown. For example, Weiss in [21] proved that Sn has a limiting normal distribution when no weight is attached to the observations below the pth sample percentile and above the qth sample percentile, p