A Concept of Generalized Order Statistics

Order statistics and record values are widely used in statistical models and inference; both describe random variables arranged in order of magnitude. Hence, the joint distribution of n order statistics or of n record values is defined on some cone of the n-dimensional Euclidean space.

Several models of ordered random variables are discussed in Section 1. In the distribution theoretical sense, these structures are included in the model of generalized order statistics which we introduce next. Section 3 contains the basic distribution theory of generalized order statistics. Representations of one- and higher dimensional marginal density functions and of the distribution function of a single generalized order statistic are derived and useful properties of them are shown.

One-dimensional marginal density functions and distribution functions have been obtained in the first chapter. In the following two chapters we derive recurrence relations and inequalities for moments of g OS’s and give corresponding characterizations of probability distributions. Hence, we should have at hand some results on the existence of such moments. We state sufficient conditions which are well known in the case of o OS’s and record values (see e.g. David 1981, Sen 1959, Lin 1987). Then we give representations for the difference of moments of successive g OS’s which are used in Chapter III and we derive moments of g OS’s with respect to special distributions. In the second section we cite results on complete function sequences which are a useful tool to obtain characterization results by means of moments.

Recurrence relations and identities for moments of o OS’s have been extensively investigated in the literature. Examples of explicit expressions for moments are given in Chapter II and, obviously, they may lead to a high numerical effort and to problems concerning the accuracy of computational results. Therefore, recurrence relations are a very useful tool. Here, relations between moments are termed ‘recurrence relations’ throughout; but we do not use this term in the strict sense. I.e., it is not necessarily possible to reconstruct a whole system of moments via some given set of moments which may be suggested by the parametrization of a certain identity. In contrast to o OS’s, recurrence relations for record values have been considered in the literature only marginally. Some relations are obtained in connection with characterizations of distributions by means of inequalities (see Chapter IV, Lin 1988a, Gajek, Gather 1991, Kamps 1991a). Another example is shown in Lin (1988b). Thus, we give some examples in the special case m = −1. In principle, the identities can be classified with respect to their validity for arbitrary or special distributions.

Bounds and approximations for moments of o OS’s are often considered in the literature. Detailed surveys on related results can be found in David (1981, Ch.4) and Arnold, Balakrishnan (1989).

In reliability theory, classes of distributions are considered which describe the life-length of components or systems. I.e., the probability of failure up to a time t ≥ 0 is modelled. There are numerous articles concerning the analysis of such families of distributions (see Barlow, Proschan 1975, 1981, Patel 1983, Basu 1988). We restrict ourselves to some articles dealing with o OS’s and record values.