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December, 1957 On a Statistic which Arises in Selection and Ranking Problems
Shanti S. Gupta, Milton Sobel
Ann. Math. Statist. 28(4): 957-967 (December, 1957). DOI: 10.1214/aoms/1177706796


The statistic $y = (x_{\lbrack p\rbrack} - x)/s_\nu$ is studied where $x_{\lbrack p\rbrack}$ is the maximum of $p$ normal independent chance variables with common mean and common unknown variance $\sigma^2, x$ is another independent normal chance variable with the same mean and the same variance $\sigma^2,$ and $s^2_\nu$ (distributed as $\sigma^2\chi^2_\nu/\nu$ with $\nu$ degrees of freedom) is an estimate of the common variance which is independent of each one of the above $p + 1$ chance variables. Several different methods are proposed and studied for computing the probability integral of $y$ and percentage points of $y$; in addition, a method for computing percentage points without first computing the probability integral of $y$ is considered. A table of (upper) percentage points of $y$ is given as Table I at the end of the paper. Applications of the statistic $y$ to several ranking and selection problems are mentioned in Section 2. Moments of $y$ are given in Section 3. In Section 7 it is shown that Table I can be used to obtain an approximation and bounds to the percentage points of a related statistic.


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Shanti S. Gupta. Milton Sobel. "On a Statistic which Arises in Selection and Ranking Problems." Ann. Math. Statist. 28 (4) 957 - 967, December, 1957.


Published: December, 1957
First available in Project Euclid: 27 April 2007

zbMATH: 0080.35504
MathSciNet: MR93846
Digital Object Identifier: 10.1214/aoms/1177706796

Rights: Copyright © 1957 Institute of Mathematical Statistics

Vol.28 • No. 4 • December, 1957
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