A Concept of Generalized Order Statistics

Order statistics and record values appear in many statistical applications and are widely used in statistical modeling and inference. Both models describe random variables arranged in order of magnitude. In addition to these well-known models, several other models of ordered random variables, known and new ones, are introduced in this book such as order statistics with non-integral sample size, sequential order statistics, k-th record values, Pfeifer' s record model, k -records from non-identical distributions and ordered random variables which arise from n truncation of distributions. These models can be effectively applied, e.g., in reliability theory. Here, an order statistic represents the life-length of some r-out-of-n-system which is an important technical structure consisting of n components. For this application, a new and more adequate model is naturally suggested. Sequential order statistics serve as a model describing certain dependencies or interactions among the system components caused by failures of components. Record values are closely connected with the occurrence times of some corresponding non-homogeneaus Poisson process and used in so~lled shock models. More flexible record models, and therefore more applicable to practical situations, are considered here. The main purpose of this book is to present a concept of generalized order statistics as a unified approach to a variety of models of ordered random variables. In the distribution theoretical sense, all of the models mentioned above are contained in the proposed model of generalized order statistics. The main purpose of this book is to present a concept of generalized order statistics as a unified approach to a variety of models of ordered random variables. Numerous related results on distributional and moment properties of order statisrics and record values are found in the literature which are deduced separately.§The concept of generalized order statistics, however, enables a common approach to structural similarities and analogies. Well known results can be subsumed, generalized, and integrated within a general framework. Therefore, the concept of generalized order statistics provides a large class of models with many interesting, important and useful properties for both the description and the analysis of practical problems.