A concept of generalized order statistics

Abstract

A form of the joint distribution of n ordered random variables is presented that enables a unified approach to a variety of models of ordered random variables, e.g. order statistics and record values. Several other models are shown. In particular, sequential order statistics are introduced as a modification of order statistics which is naturally suggested by a statistical application in reliability theory. In the distribution theoretical sense, all of these models of ordered random variables are contained in the proposed concept of generalized order statistics. Numerous related results on distributional and moment properties of ordinary order statistics and record values are found in the literature which are deduced separately. Generalized order statistics, however, provide a suitable approach to explain these similarities and analogies in the two models and to generalize related results. Through integration of known properties the structure of the embedded models becomes clearer. On the other hand, we obtain the validity of these properties and their generalizations for generalized order statistics, and hence for different models of ordered random variables. In the present paper we develop the distribution theory for generalized order statistics. Representations for the one-, two- and higher-dimensional marginal densities and a form of the one-dimensional marginal distribution functions are given as well as recurrence relations for marginal densities and distribution functions. Moreover, we give representations for moments and differences of moments of generalized order statistics, sufficient conditions for the existence of moments, and we show some explicit expressions for the moments of generalized order statistics based on power function, Pareto and Weibull distributions.