Preservation of some partial orderings under the formation of coherent systems

The reversed (backward) hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their reversed hazard rate functions. In this paper, we have given some sufficient conditions under which the ordering between the components with respect to the reversed hazard rate is preserved under the formation of coherent systems. We have also shown that these sufficient conditions are satisfied by k-out-of-n systems. Both the cases when components are identically distributed and not necessarily identically distributed are discussed. Some results for likelihood ratio order are also obtained. The parallel (series) systems of not necessarily iid components have been characterized by means of a relationship between the reversed hazard rate (hazard rate) function of the system and the reversed hazard rate (hazard rate) functions of the components.