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

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
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