Some Stochastic Ordering Problems And Their Applications

Stochastic ordering is mainly used to discuss the size or sequence between two random variables,it is a partial order relationship which comparing the size of random variables under the sense of the probability distribution.Random variables are first compared by comparing their expectations and variances.Although it was widely used in many fields,it was soon found that the expectation and variance could not fully reflect the information of random variables,especially the relationship between the size of random variables.Therefore,the concept of stochastic ordering is introduced to make up for the deficiency of comparing random variables only by numerical features.Lehmann first proposed the concept of stochastic ordering in 1955,then the study of stochastic ordering has been paid more and more attention by statisticians.Stochastic ordering theory has important applications in many fields,such as reliability,risk management,life test and so on.It has become an important branch of statistics.As a theoretical and widely used tool,order statistics playing an important role in stochastic ordering.We mainly study the common stochastic orderings of the largest and smallest order statistics under the different independent samples based on Rayleigh distribution,Weibull distribution and Logistic distribution in this thesis.Firstly,we discussed stochastic ordering for the Rayleigh distribution.A counterexample is given to illustrate the likelihood ratio ordering of the maximum order statistic of Rayleigh distribution that cannot be established by majorization,then we consider the likelihood ratio ordering under the multiple-outlier Rayleigh distribution,and the conditions for satisfying the parameters are further generalized.Finally,a counterexample is given to show that the condition cannot be generalized to the general weak majorization.Secondly,we discussed stochastic ordering for n independent random variables with different distributions in Weibull distribution,The sequence is obtained when the shape parameter is between 0-1,and the scale parameter establishes the usual stochastic ordering,the hazard rate ordering and the likelihood ratio ordering of the Weibull distribution minimum order statistics by majorization;On this basis we discussed when the two sets of random variables are different in shape parameters,the scale parameters satisfy the likelihood ratio ordering of the maximum order statistic of the multiple-outlier Weibull distribution.Finally,we discussed the stochastic ordering problem of order statistics in Logistic distribution.According to majorization,we establish the usual stochastic ordering and the hazard rate ordering of maximum order statistics and minimum order statistics of two sets of random samples in Logistic distribution respectively.For the likelihood ratio ordering,we mainly consider the case of two sets of the multiple-outlier Logistic distribution.