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June, 1953 On the Asymptotic Normality of Certain Rank Order Statistics
Meyer Dwass
Ann. Math. Statist. 24(2): 303-306 (June, 1953). DOI: 10.1214/aoms/1177729038


Let $(R_1, \cdots, R_N)$ be a random vector which takes on each of the $N!$ permutations of the numbers $(1, \cdots, N)$ with equal probability, $1/N!$. Sufficient conditions are given for the asymptotic normality of $S_N = \sum^N_{i=1}a_{Ni}b_{NR_i}$, where $(a_{N1}, \cdots, a_{NN}), (b_{N1}, \cdots, b_{NN})$ are two sets of real numbers given for every $N$. These sufficient conditions are apparently quite different from those given by Wald and Wolfowitz [9] and extended by various writers [4, 7]. In some situations the conditions given here may be easier to apply than those given previously. The most general conditions available to date appear to be those of Hoeffding [4]. In the examples below, however, is given a case of an $S_N$ which does not satisfy the conditions required by Hoeffding's theorem but which is asymptotically normal by our results.


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Meyer Dwass. "On the Asymptotic Normality of Certain Rank Order Statistics." Ann. Math. Statist. 24 (2) 303 - 306, June, 1953.


Published: June, 1953
First available in Project Euclid: 28 April 2007

zbMATH: 0053.10302
MathSciNet: MR55628
Digital Object Identifier: 10.1214/aoms/1177729038

Rights: Copyright © 1953 Institute of Mathematical Statistics

Vol.24 • No. 2 • June, 1953
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