Generalized densities of order statistics

Abstract

Let X1 ,⋯, Xn be independent identically distributed random variables with distribution F. We derive expressions for generalized joint 'densities' of order statistics of X1 ,⋯, Xn, for arbitrary distributions F, in terms of Radon-Nikodym derivatives with respect to product measures based on F. We then give formulae for conditional distributions of order statistics and use them to derive results concerning Markov properties of order statistics, formulae for distributions of trimmed sums, and other useful representations. Our approach leads to simple and natural expressions which appear not to have been given before.