Limit Theory for Joint Generalized Order Statistics

Abstract

In Kamps [7] generalized order statistics (gos) have been introduced as a unifying theme for several models of ascendingly ordered random variables (rv’s). The main aim of this paper is to study the limit joint distribution function (df) of any two statistics in a wide subclass of the gos model known as m-gos. This subclass contains many important practical models of gos such as ordinary order statistics (oos), order statistics with non-integer sample size, and sequential order statistics (sos). The limit df’s of lower-lower extreme, upper-upper extreme, lower-upper extreme, centralcentral and lower-lower intermediate m-gos are obtained. It is revealed that the convergence of the marginals m-gos implies the convergence of the joint df. Moreover, the conditions, under which the asymptotic independence between the two marginals occurs, are derived.