Bivariate Generalized Order Statistics: Asymptotic Theory of Extreme, Central and Intermediate

In Kamps (1995) generalized order statistics (GOS) have been introduced as a unifying theme for several models of ascendingly ordered random variables (rv's). Following Kamps, Burkschat et al. (2003) have introduced the concept of dual generalized order statistics (DGOS) to unify several models that produce ordered rv's. The main aim of this book is to study the limit joint distribution function (df) of any two statistics in a wide subclass of the GOS and DGOS models known as m-GOS and m-DGOS respectively. This subclass contains many important practical models such as ordinary order statistics, order statistics with non-integer sample size, sequential order statistics and upper and lower record values. The limit df's of lower-lower extreme, upper-upper extreme, lower-upper extreme, central-central and lower-lower intermediate m-GOS and m-DGOS are obtained. It is revealed that the convergence of the marginals m-GOS and m-DGOS implies the convergence of the joint df. Moreover, the conditions, under which the asymptotic independence between the two marginals occurs, are derived.