Stochastic Properties Of Largest Order Statistics From Heterogeneous Gamma Samples

In this thesis, we investigate the ordering properties of largest order statistics arising from heterogeneous gamma samples with both different shape and scale parameters. First of all, it is proved that the weak supermajorization order be-tween the scale parameter vectors together with the weak submajorization order between the shape parameter vectors imply the reversed hazard rate ordering for the maximum order statistics arising from two sets of independent heterogeneous gamma variables. Also, we show that the usual stochastic ordering holds under the p-larger order between the scale parameter vectors and the weak submajoriza-tion order between the shape parameter vectors. For the case when n= 2, we extend the above results to the likelihood ratio ordering and hazard rate ordering, respectively. Besides, we also study the skewness according to the star ordering by providing a very general sufficient condition, using which some useful conse-quences can be obtained. Some numerical examples and practical applications in system reliability, auction theory and minimal repairs are provided to expli-cate the results. The new results established here strengthen and generalize some known results in the literature, which are of great importance both in theory and reality.