Distribution of the Maximum of Concomitants of Selected Order Statistics

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.