Ordinal Variables Model The Dependency Structure

Dependence structures of order statistics are becoming increasingly important in proba-bility,statistics,reliability theory,survival analysis and other fields.This thesis focuses on the nonnegativity of covariances of functions of two adjacent ordered ordered random variables,and the dependence properties of general spacing vector of ordinary order statistics.Dependence structure of a class of symmetric distributions is also considered.Firstly,We investigate conditions by a unified method under which the covariances of functions of two adjacent ordered random variables are nonnegative.Several main structural results are presented and then are applied to several kinds of ordered random variable,such as delayed record values,continuous and discrete(?)-spherical order statistics,epoch times of mixed Poisson processes,generalized order statistics,discrete weak record values,and epoch times of modified geometric processes,concomitants of order statistics and record values.These applications extend the main results for ordinary order statistics in Qi(1994)and for usual record values in Nagaraja and Nevzorov(1996).Secondly,for order statistics of independent identically distributed random variables,the MTP₂ and strong-MRR₂ properties of general spacing vectors corresponding to one random sample are considered.We also investigate the multivariate likelihood ratio orderings be-tween general spacing vectors from two samples.For order statistics of independent but non-identically distributed random variables,we discuss conditional orderings of order statistics, extending the main results in Dubhashi and H(?)gstrom(2007),and Hu and Chen(2008).Finally,dependence structure of a class of symmetric distributions is considered.We investigate conditions under which the underlying random variables are MTP₂,and the mul-tivariate likelihood ratio ordering between two samples can be obtained.We also establish the nonnegativity of the covariance between functions of two adjacent order statistics of the underlying random variables.