Generalized Order Statistics

AbstractOrder 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 ascending order of magnitude. Generalized order statistics provide a unified approach to a variety of models of ordered random variables with different interpretations, such as ordinary order statistics, sequential order statistics, progressively type II censored order statistics, record values,kth record values, and Pfeifer's records. These models can be effectively applied, for example, in reliability theory. Here, an order statistic represents the life length of somek‐out‐of‐nsystem, which is an important technical structure. For this application, sequential order statistics serve to describe influences of failures on the life‐length distributions of the remaining components. Record values are used in shock models, and they are closely connected with the occurrence times of some corresponding nonhomogeneous Poisson process, with some minimal repair scheme, and with the relevation transform. The distribution theory of generalized order statistics is established along the lines of ordinary order statistics. By this, we find that many well‐known properties of order statistics and record values are also valid for generalized order statistics. The results on generalized order statistics concern their distribution theory, relations and bounds for moments, characterizations of distributions, preservation of aging properties and stochastic ordering, extreme‐value theory, and statistical inference with respect to model and distribution parameters.