The Efficiency of the Mean Moving Range

Abstract

In studying the variation of a variable subject to erratic trend effects, it is customary to employ as a measure of variation a statistic that eliminates most of such effects. It is shown in this paper that the statistic $w = \sum^{n - 1}_1 \|x_{i+1} - x_i\| \sqrt{\pi}/2(n - 1)$ is nearly as efficient as the statistic $\delta^2 = \sum^{n - 1}_1 (x_{i+1} - x_i)^2/(n - 1)$ that is customarily employed. The asymptotic variance of w is obtained by integration techniques; the proof of the asymptotic normality of w is based upon a theorem of S. Bernstein on the asymptotic distribution of sums of dependent variables. The method of proof is sufficiently general to prove the asymptotic normality of w, and of $\delta^2$, for $x$ having a distribution for which the third absolute moment exists.