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December, 1959 Contributions to the Theory of Rank Order Statistics--The One-Sample Case
I. Richard Savage
Ann. Math. Statist. 30(4): 1018-1023 (December, 1959). DOI: 10.1214/aoms/1177706084


The one-sample problem is considered using techniques developed earlier [2], [3]. Let $Z = (Z_1, \cdots, Z_N)$ be a random vector with $Z_i = 1(0)$ if the $i$th smallest in absolute value in a sample of $N$ from the density $f(x)$ is positive (negative). Then $$P(Z = z) = N! \int_{\cdots_{0\leqq y_1\leqq\cdots\leqq yN\leqq\infty}}\int \prod_{i=1}^N \lbrack f^{1-z_i} (-y_i)f^{z_i}(y_i) dy_i\rbrack$$ Conditions are found implying $P(Z = z) > P(Z = z')$ where $z$ is derived from $z'$ by replacing a 0 by a 1, or interchanging a 0 and 1 in $z'$ by moving the 1 to the right. These conditions are met by the normal and other distributions. The results are useful in finding good tests of such null hypotheses as $X_1, \cdots, X_N$ are independently and identically distributed symmetrically about zero against such alternatives as slippage to the right. The Wilcoxon one sample signed rank test is a typical nonparametric procedure used under these conditions [4].


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I. Richard Savage. "Contributions to the Theory of Rank Order Statistics--The One-Sample Case." Ann. Math. Statist. 30 (4) 1018 - 1023, December, 1959.


Published: December, 1959
First available in Project Euclid: 27 April 2007

zbMATH: 0116.37404
MathSciNet: MR109403
Digital Object Identifier: 10.1214/aoms/1177706084

Rights: Copyright © 1959 Institute of Mathematical Statistics

Vol.30 • No. 4 • December, 1959
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