## The Annals of Statistics

- Ann. Statist.
- Volume 5, Number 1 (1977), 216-223.

### Distribution and Expected Value of the Rank of a Concomitant of an Order Statistic

H. A. David, M. J. O'Connell, and S. S. Yang

#### Abstract

Let $(X_i, Y_i)$ be $n$ independent rv's having a common bivariate distribution. When the $X_i$ are arranged in nondecreasing order as the order statistics $X_{r:n} (r = 1,2,\cdots, n)$, the $Y$-variate $Y_{\lbrack r:n\rbrack}$ paired with $X_{r:n}$ is termed the concomitant of the $r$th order statistic. The small-sample theory of the distribution and expected value of the rank $R_{r:n}$ of $Y_{\lbrack r:n\rbrack}$ is studied. In the special case of bivariate normality an illustrative table of the probability distribution of $R_{r,n}$ is given. A more extensive table of $E(R_{r,n})$ is also provided and it is found that asymptotic results require comparatively small finite-sample corrections even for modest values of $n$. Some applications are briefly indicated.

#### Article information

**Source**

Ann. Statist., Volume 5, Number 1 (1977), 216-223.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343756

**Digital Object Identifier**

doi:10.1214/aos/1176343756

**Mathematical Reviews number (MathSciNet)**

MR445673

**Zentralblatt MATH identifier**

0356.62040

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G30: Order statistics; empirical distribution functions

Secondary: 62F07: Ranking and selection

**Keywords**

Order statistics concomitants ranking selection bivariate normal tables

#### Citation

David, H. A.; O'Connell, M. J.; Yang, S. S. Distribution and Expected Value of the Rank of a Concomitant of an Order Statistic. Ann. Statist. 5 (1977), no. 1, 216--223. doi:10.1214/aos/1176343756. https://projecteuclid.org/euclid.aos/1176343756