A STUDY OF ASYMPTOTIC DISTRIBUTIONS OF CONCOMITANTS OF CERTAIN ORDER STATISTICS

• Published in: Statistica Sinica 1999-07, Vol.9 (3), p.811-830
• Main Authors: Chu, Shu-Jane, Huang, Wen-Jang, Chen, Hung
• Format: Article
• Language: English
• Subjects: Applied mathematics; Conditional convergence; Conditional probabilities; Distribution functions; Infinity; Integers; Random sampling; Random variables; Sampling distributions; Statistics
• ISSN: 1017-0405; 1996-8507

Consider a random sample of size n from an absolutely continuous bivariate distribution of (X, Y). Let Xi:n denote the ith order statistic of the X sample values and Y[i:n] its concomitant, the Y-value associated with Xi:n. In this article we are interested in the asymptotic behavior of some functionals of concomitants. Given two increasing integer sequences {rn, n ≥ 1} and {sn, n ≥ 1} with 1 ≤ rn ≤ sn ≤ 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 limn→∞(n − rn) < ∞ or limn→∞ rn/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).