## The Annals of Statistics

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

### Functions Decreasing in Transposition and Their Applications in Ranking Problems

Myles Hollander, Frank Proschan, and Jayaram Sethuraman

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343895

**Digital Object Identifier**

doi:10.1214/aos/1176343895

**Mathematical Reviews number (MathSciNet)**

MR488423

**Zentralblatt MATH identifier**

0356.62043

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H10: Distribution of statistics

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176343895