Stochastic Comparisons In Frailty Models And Order Statistics

In this thesis, we have a thorough study on stochastic comparisons in frailty models as well as order statistics from independent heterogeneous exponential random variables.As a particularly useful tool for handing heterogeneity left unexplained by observed covariates, during recent decades, frailty models have been widely applied in such as epidemiology, demography and other areas related to survival analysis. In the first part of this thesis, we focus on stochastic comparisons in several frailty models.Firstly, in classical frailty models introduced by Vaupel et al. (1979), we conduct stochastic comparisons of the overall population and an individual with specific frailty, and establish equivalent characterizations for some well known stochastic orders between them in terms of the frailty and moments of the frailty variable. Upon these results we further compare the high risk groups with the overall population in risk assessment.Secondly, in the general frailty model due to Gupta and Gupta (2009), the overall populations arising from different choices of the distribution of the frailty variable are compared. In particular, we correct a false result of Gupta and Gupta (2009).Thirdly, we study the competing system between two groups of observations, each of them follows multivariate frailty models. Calculation formulas for some evaluation indices of the system are derived, and how the variation of the frailty vector has an impact on the performance of this system is investigated as well.Lastly, based on univariate proportional reversed hazard rate frailty model, we have a further study on two kinds of frailty models, one is the univariate general frailty model in terms of the reversed hazard rate, the other is the multivariate proportional reversed hazard rate frailty model. For the former, some monotone properties of the overall pop-ulation (mixture) reversed hazard rate and the population inactivity time are discussed. Besides, a sufficient condition for the preservation of DRHR under the mixture is derived as well. For the latter, we investigate how some of the well known stochastic orders of the frailty vector and some ageing properties of the baseline are translated into those of the overall population. Also, the stochastic monotone properties of the overall population vector with respect to the frailty vector are discussed. And the overall population reversed hazard rate is discussed as well.Order statistics from heterogeneous observations play an important role in various areas related to applied probability and statistics such as reliability theory, survival anal-ysis, operations research and actuarial sciences. In the second part, we conduct stochastic comparisons on order statistics from independent heterogeneous exponential random vari-ables.Firstly, we establish equivalent characterizations in terms of parameters for the right spread ordering between the second order statistic (Fail-Safe system) from nonidentical independent exponential observations and that from i.i.d. exponential ones.Secondly, we build a sufficient condition for the hazard rate ordering between lifetimes of parallel systems with two independent but nonidentical exponential components, this also serves as an answer to the open problem of Joo and Mi (2010). Moreover, this result is extended to the case with proportional hazard rates. Some comparisons on lifetimes of such systems with general components are also obtained.