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December, 1946 The Efficiency of the Mean Moving Range
Paul G. Hoel
Ann. Math. Statist. 17(4): 475-482 (December, 1946). DOI: 10.1214/aoms/1177730886


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.


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Paul G. Hoel. "The Efficiency of the Mean Moving Range." Ann. Math. Statist. 17 (4) 475 - 482, December, 1946.


Published: December, 1946
First available in Project Euclid: 28 April 2007

zbMATH: 0063.02036
MathSciNet: MR22055
Digital Object Identifier: 10.1214/aoms/1177730886

Rights: Copyright © 1946 Institute of Mathematical Statistics


Vol.17 • No. 4 • December, 1946
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