Linear Functions of Order Statistics with Smooth Weight Functions

This paper considers linear functions of order statistics of the form Sₙ = n⁻¹ ∑ J(i/(n + 1))X₍ᵢ₎. The main results are that Sₙ is asymptotically normal if the second moment of the population is finite and J is bounded and continuous a.e. F⁻¹, and that this first result continues to hold even if the unordered observations are not identically distributed. The moment condition can be discarded if J trims the extremes. In addition, asymptotic formulas for the mean and variance of Sₙ are given for both the identically and non-identically distributed cases. All of the theorems of this paper apply to discrete populations, continuous populations, and grouped data, and the conditions on J are easily checked (and are satisfied by most robust statistics of the form Sₙ). Finally, a number of applications are given, including the trimmed mean and Gini's mean difference, and an example is presented which shows that Sₙ may not be asymptotically normal if J is discontinuous.