Joint distribution of maxima of concomitants of subsets of order statistics. (English) Zbl 0836.62038

Joshi, S. N.; Nagaraja, H. N.

Bernoulli 1, No. 3, 245-255 (1995).

Summary: Let \((X_{i : n},\;Y_{[i : n]})\), \(1 \leq i \leq n\), denote the \(n\) pairs obtained by ordering a random sample of size \(n\) from an absolutely continuous bivariate population on the basis of \(X\) sample values. Here \(Y_{[i : n]}\) is called the concomitant of the \(i\)th order statistic. For \(1 \leq k \leq n\), let \(V_1 = \max \{Y_{[n - k + 1 : n]}, \dots, Y_{[n : n]}\}\), and \(V_2 = \max \{Y_{[1 : n]}, \dots, Y_{[n - k : n]}\}\).
We discuss the finite-sample and asymptotic joint distribution of \((V_1, V_2)\). The asymptotic results are obtained when \(k = [np]\), \(0 < p < 1\), and when \(k\) is held fixed, as \(n \to \infty\). We apply our results to the bivariate normal population and indicate how they can be used to determine \(k\) such that \(V_1\) is close to \(Y_{n : n}\), the maximum of the values of \(Y\) in the sample.

MSC:

62G30	Order statistics; empirical distribution functions
62E20	Asymptotic distribution theory in statistics

Keywords:

concomitants of order statistics; convergence in distribution; extreme values maximum; quantiles; asymptotic joint distribution; bivariate normal population