Statistica Sinica 9(1999), 811-830

A STUDY OF ASYMPTOTIC DISTRIBUTIONS OF

CONCOMITANTS OF CERTAIN ORDER STATISTICS

Shu-Jane Chu, Wen-Jang Huang and Hung Chen$^*$

National Sun Yat-sen University and $^*$National Taiwan University

Abstract: Consider a random sample of size $n$ from an absolutely continuous bivariate distribution of $(X,Y)$. Let $X_{i:n}$ denote the $i$th order statistic of the $X$ sample values and $Y_{[i:n]}$ its concomitant, the $Y$-value associated with $X_{i:n}$. In this article we are interested in the asymptotic behavior of some functionals of concomitants. Given two increasing integer sequences $\{r_n, n\geq 1\}$ and $\{s_n, n\geq 1\}$ with $1\leq r_n \leq s_n \leq n$, let $V_{r_n,s_n,n}= \max (Y_{[r_n:n]},Y_{[r_n+1:n]},\ldots, Y_{[s_n:n]})$ and $W_{r_n,s_n,n}= \min (Y_{[r_n:n]},Y_{[r_n+1:n]}, \ldots,Y_{[s_n:n]})$. We investigate the limiting distributions of $V_{r_n,s_n,n}$, $W_{r_n,s_n,n}$, $R_{r_n,s_n,n}= V_{r_n,s_n,n} - W_{r_n,s_n,n}$, and $M_{r_n,s_n,n}= (V_{r_n,s_n,n} +W_{r_n,s_n,n})/2$ when $\lim_{n \rightarrow \infty}(n-r_n) <\infty$ or $\lim_{n\rightarrow \infty} r_n/n = p$ for $0 < p < 1$. The statistics $R_{r_n,s_n,n}$ and $M_{r_n,s_n,n}$ can be viewed as range and midrange, respectively. Our results generalize those obtained in Nagaraja and David (1994). We also use these results to investigate the problem of locating the maximum of a nonparametric regression function as discussed in Chen, Huang and Huang (1996).

Key words and phrases: Concomitants of order statistics, regression model.