Stochastic Comparisons Of Order Statistics From Several Common Distributions

Order statistics play an important role in statistical inference, reliability theory, life testings, application of probability and many other areas.In reliability theory, k-out-of-n systems consisting of n components work if and only if at least k components are working, or equivalent to say, k-out-of-n systems consisting of n components work if and only if at least n-k components are failure. The survival function of a k-out-of-n system is the same as that of the (n-k+1)th order statistics of a set of n random vari-ables. In particular, the parallel systems are 1-out-of-n systems, the series systems are n-out-of-n. Thus, the study of lifetimes of parallel systems is equivalent to the study of the largest order statistics, and the study of lifetimes of series systems is equivalent to the study of the smallest order statistics. Order statistics have been extensively inves-tigated in the case when the observations are independent and identically distributed. However, observations are independent and nonidentical distributed or nonindepen-dent and nonidentical distributed in many practical situations. In this paper, we mainly investigate two sets consisting of independent and nonidentical distributed or noninde-pendent and nonidentical distributed random variables. The main work and conclusions are as follows:(1)、We establish the stochastic comparisons of the largest and the smallest or-der statistics with respect to majorization based on two independent Gaussian vectors, but the components in the vector are dependent respectively. We consider two cases as follows. One case is two Gaussian random vectors have equal mean, sharing the same correlation coefficient for all components and the vector being constituted by the reciprocal of each component’s standard deviation satisfying majorization; the other case is two Gaussian random vectors with equal covariance matrix and the mean vector satisfying majorization.(2)、Firstly, we provide some comparisons, for example, the convex transform order, the dispersive order and the reserved hazard rate order, on the lifetimes of series systems arising from n independent heterogeneous Weibull random variables and n i.i.d. Weibull random variables. Secondly, we compare the lifetime of two series systems with n independent Weibull components with respect to the reversed hazard rate order.(3)、Firstly, we discuss ordering properties of lifetimes of series systems with two independent heterogeneous Weibull components in terms of dispersive order and star order. We give some sufficient conditions imply the above two orders between lifetimes of series systems. Secondly, we give some sufficient conditions, based on the scale parameters, on the lifetimes of series systems arising from two independent heterogeneous Weibull random variables in terms of the right spread order.(4)、We study some stochastic comparisons of parallel systems with n indepen-dent exponentiated Weibull components. The results are the generalization of the re-sults of exponential or Weibull distributions.(5)、We obtain the usual multivariate stochastic ordering between vectors of order statistics from two sets of proportional reversed hazard rate models under the condition of the proportional reversed hazard rate parameters satisfying majorization.The innovations of the methodologies and results in this dissertation are described as following. Firstly, we establish the vector being constituted by the reciprocal of each component’s standard deviation or the mean vector satisfying majorization based on two independent Gaussian vectors, but the components in the vector are dependent re-spectively. So, we give the stochastic comparisons of the largest and the smallest order statistics according to the usual order. The results generalize the classical Slepian’s inequality. Secondly, we study stochastic comparisons of lifetimes of series systems with two independent heterogeneous Weibull components in terms of the convex trans-form order, the dispersive order and the reserved hazard rate order. Thirdly, We provide some sufficient conditions, based on the scale parameter, on the stochastic comparisons of lifetimes of series systems arising from two independent heterogeneous Weibull ran-dom variables. Fourthly, we give some stochastic comparisons of lifetimes of parallel systems with exponentiated Weibull components in terms of the usual order, the re-served hazard rate order, the dispersive order and the likelihood ratio order. At last, we give the usual multivariate stochastic ordering between vectors of order statistics from two sets of proportional reversed hazard rate models by introducing Beta variables and combining the usual multivariate stochastic ordering between vectors of order statistics from two sets of proportional hazard rate models.