The Annals of Statistics

Distribution of the Maximum of Concomitants of Selected Order Statistics

H. N. Nagaraja and H. A. David

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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

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

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62E20: Asymptotic distribution theory
Secondary: 62G30: Order statistics; empirical distribution functions 62E15: Exact distribution theory

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


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.

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