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July, 1977 Functions Decreasing in Transposition and Their Applications in Ranking Problems
Myles Hollander, Frank Proschan, Jayaram Sethuraman
Ann. Statist. 5(4): 722-733 (July, 1977). DOI: 10.1214/aos/1176343895

Abstract

Let $\mathbf{\lambda} = (\lambda_1, \cdots, \lambda_n), \lambda_1 \leqq \cdots \leqq \lambda_n$, and $\mathbf{x} = (x_1, \cdots, x_n)$. A function $g(\mathbf{\lambda, x})$ is said to be decreasing in transposition (DT) if (i) $g$ is unchanged when the same permutation is applied to $\mathbf{\lambda}$ and to $\mathbf{x}$, and (ii) $g(\mathbf{\lambda, x}) \geqq g(\mathbf{\lambda, x}')$ whenever $\mathbf{x}'$ and $\mathbf{x}$ differ in two coordinates only, say $i$ and $j, (x_i - x_j) \cdot (i - j) \geqq 0$, and $x_i' = x_j, x_j' = x_i$. The DT class of functions includes as special cases other well-known classes of functions such as Schur functions, totally positive functions of order two, and positive set functions, all of which are useful in many areas including stochastic comparisons. Many well-known multivariate densities have the DT property. This paper develops many of the basic properties of DT functions, derives their preservation properties under mixtures, compositions, integral transformations, etc. A number of applications are then made to problems involving rank statistics.

Citation

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Myles Hollander. Frank Proschan. Jayaram Sethuraman. "Functions Decreasing in Transposition and Their Applications in Ranking Problems." Ann. Statist. 5 (4) 722 - 733, July, 1977. https://doi.org/10.1214/aos/1176343895

Information

Published: July, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0356.62043
MathSciNet: MR488423
Digital Object Identifier: 10.1214/aos/1176343895

Subjects:
Primary: 62H10
Secondary: 62E15 , 62F07 , 62G99

Keywords: decreasing in transposition , positive set functions , preservation properties , rank statistics , Schur functions , totally positive functions

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 4 • July, 1977
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