## The Annals of Statistics

- Ann. Statist.
- Volume 22, Number 1 (1994), 478-494.

### Distribution of the Maximum of Concomitants of Selected Order Statistics

H. N. Nagaraja and H. A. David

#### Abstract

For a random sample of size $n$ from an absolutely continuous bivariate population $(X, Y)$, let $X_{i:n}$ denote the $i$th order statistic of the $X$ sample values. The $Y$-value associated with $X_{i:n}$ is denoted by $Y_{\lbrack i:n\rbrack}$ and is called the concomitant of the $i$th order statistic. For $1 \leq k \leq n$, let $V_{k,n} = \max(Y_{\lbrack n - k + 1: n\rbrack},\ldots,Y_{\lbrack n: n\rbrack})$. In this paper, we discuss the finite-sample and the asymptotic distributions of $V_{k,n}$. We investigate the limit distribution of $V_{k,n}$ as $n \rightarrow \infty$, when $k$ is held fixed and when $k = \lbrack np\rbrack, 0 < p < 1$. In both cases we obtain simple sufficient conditions and determine the associated norming constants. We apply our results to some interesting situations, including the bivariate normal population and the simple linear regression model.

#### Article information

**Source**

Ann. Statist., Volume 22, Number 1 (1994), 478-494.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176325380

**Digital Object Identifier**

doi:10.1214/aos/1176325380

**Mathematical Reviews number (MathSciNet)**

MR1272095

**Zentralblatt MATH identifier**

0795.62010

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62G30: Order statistics; empirical distribution functions 62E15: Exact distribution theory

**Keywords**

Maximum concomitants of order statistics weak convergence tail equivalence bivariate normal distribution simple linear regression model

#### Citation

Nagaraja, H. N.; David, H. A. Distribution of the Maximum of Concomitants of Selected Order Statistics. Ann. Statist. 22 (1994), no. 1, 478--494. doi:10.1214/aos/1176325380. https://projecteuclid.org/euclid.aos/1176325380