The Annals of Statistics

Distribution of the Maximum of Concomitants of Selected Order Statistics

H. N. Nagaraja and H. A. David

Full-text: Open access

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


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