Limit theorems for univariate and bivariate order statistics with variable ranks

Abstract

In this work, we investigate the asymptotic behaviour of the intermediate and central order statistics of a bivariate data by using the Reduced Ordering Principle (R-ordering). When, the sup-norm is used, we reveal the interrelation between the R-ordering principle and Marginal-Ordering Principle (M-ordering). The asymptotic behaviour of the intermediate and central bivariate order statistics based on sup-norms is completely determined. Besides, we highlight the role of the normalizing constants in characterizing the limit types of order statistics and new results concerning the intermediate order statistics are given.

Keywords:

• Weak convergence
• order statistics with variable rank
• reduced-ordering principle
• marginal-ordering principle
• sup-norm

AMS 2000 Subject Classifications:

• Primary 62G30
• 62E20
• Secondary 62H10

Acknowledgments

The authors are very grateful to the Editor in Chief, Professor Meister, and anonymous referees for their insightful review that enabled the authors to improve the paper significantly.

Disclosure statement

No potential conflict of interest was reported by the authors.