Stochastic Comparisons Of Order Statistics From Several Distributions

Order statistics play an important role in statistical inference, reliability theory, life testing, application of probability and many other areas. As a useful tool, such as the random sequence in its random comparison plays an extremely important role. This paper aims to study in the two parameter exponential distribution, distribution and Laplace distribution and two parameter Rayleigh distribution model between the order statistics random comparison, using the concept of Schur functions and the nature of the order, were found in the four distribution parameters satisfy some conditions, and under these conditions are of the order statistics of a sample in a different sense of stochastic comparison.For the two groups of samples from the two parameter exponential distribution, when the parameters are satisfied with??1,…,?n??2 m??1^*,…,?n^*?, the minimum order statistic is in the sense of random order, failure rate order and likelihood ratio has X_1:n??,???_stX_1:n^*??,?^*?,X_1:n??,???hrX_1:n^*??,?^*??X_1:n??,???_lrX_1:n^*??,?^*?; When the parameters are satisfied with ??1,?2,…,?n??m??1^*,?2^*,…,?n^*?, the minimum order statistic has the meaning of the normal random order, the failure rate order, the reverse failure rate order and the likelihood ratio of the two sets of non homogeneous samples. X_1:n?_stX_1:n^*,X_1:n?hrX_1:n^*,X_1:n?rhX_1:n^*?X_1:n?_lrX_1:n^* When the order relation is satisfied with ?1/?1,1/?2,…,1/?n??m(1/?1^*,1/?2^*,…,1/?n,the maximum order statistic is in the sense of the reverse order X_n:n?rhX_n:n^*; For two sets of samples from Laplace distribution, when the parameters are satisfied with ??1,…,?n??2m??1^*,…,?n^*?, the minimum order statistic is in the sense of the normal random order, the failure rate order and the likelihood ratio order X_1:n??_i,?_i??_stX_1:n^*??_i^*,?_i?,X_1:n??_i,?_i??hrX_1:n^*??_i^*,?_i??X_1:n??_i,?_i??_lrX_1:n^*??_i^*,?_i?, The maximum order statistic is in the sense of the usual random order, the reverse failure rate order and the likelihood ratio order X_n:n??,???_stX_n:n^*??^*,??,X_n:n??_i,?_i??rhX_n:n^*??_i^*.?_i??X_n:n??_i,?_i??_lrX_n:n^*??_i^*,?_i?;For two sets of samples from the two parameter distribution, the maximum order statistics are obtained when the parameters satisfy with ??p?^* and ??p?^*, the order relation and the maximum order statistics are obtained X_n:n??,???_stX_n:n^*??^*,?^*?,X_n:n??,???rhX_n:n^*??^*,?^*??X_n:n??,???_lrX_n:n^*??^*,?^*?.