The Annals of Statistics

Functions Decreasing in Transposition and Their Applications in Ranking Problems

Myles Hollander, Frank Proschan, and Jayaram Sethuraman

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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.

Article information

Ann. Statist., Volume 5, Number 4 (1977), 722-733.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62H10: Distribution of statistics
Secondary: 62E15: Exact distribution theory 62F07: Ranking and selection 62G99: None of the above, but in this section

Decreasing in transposition rank statistics preservation properties Schur functions totally positive functions positive set functions


Hollander, Myles; Proschan, Frank; Sethuraman, Jayaram. Functions Decreasing in Transposition and Their Applications in Ranking Problems. Ann. Statist. 5 (1977), no. 4, 722--733. doi:10.1214/aos/1176343895.

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