Generalized Order Statistics Of Random Comparison

Generalized order statistics (GOS's, for short) are a subclass of sequential order statistics, and contain a variety of models of ordered random variables often used in probability and statistics, e.g., ordinary order statistics, record values, k-record values, Pfeifer's record model, progressive type II censored order statistics, order statistics under multivariate imperfect repair, and so on. GOS's provide a suitable approach to explain these similarities and analogies in those models and to generalize related results. Through integration of known properties the structure of the embedded models becomes clearer. This thesis focuses on stochastic comparisons of GOS's and its spacings in several stochastic orders. We have obtained many new properties of GOS's, which can be used in other special models of ordered random variables.For the particular case of GOS's - ordinary order statistics, we establish some further results on univariate stochastic comparisons for m-spacings in the likelihood ratio and the hazard rate orders.We investigate some further univariate stochastic comparisons of GOS's under the more general assumptions on the parameters of GOS's. For GOS's based on the same distribution, we present the preservation properties of the likelihood ratio, hazard rate and reversed hazard rate orders; For GOS's based on two different distributions, we obtain the preservation properties of the hazard rate and the dispersive orders.For GOS's from one sample, we establish several stochastic comparisons of general p-spacings in the likelihood ratio and the hazard rate orders, and moreover, preservation properties of the log-convexity and logconcavity of p-spacings are given; For GOS's from two samples, we establish the likelihood ratio ordering of general p-spacings and the hazard rate and the dispersive orderings of normalizing simple spacings.We finally establish some results on multivariate stochastic comparisons of one-sample sequential order statistics in the multivariate likelihood ratio, the hazard rate, and the usual stochastic orders, so we can conclude some further results on univariate stochastic comparisons of sequential order statistics in the likelihood ratio and the usual stochastic orders. Furthermore, based on the interrelationship between sequential order statistics and epoch times of nonhomogeneous pure birth (NHPB) processes, we obtain stochastic comparisons of two-sample sequential order statistics in the multivariate likelihood ratio, the hazard rate, and the usual stochastic orders. Applications in some special models of sequential order statistics and in the NHPB process are also presented.