A general composition theorem and its applications to certain partial orderings of distributions

A composition theorem for functions obeying certain positive ordering is proved. The novelty of the present version is that unlike earlier results which assume both components of the composition to be distributions or survival functions, one of the components is allowed to be negative and unbounded. The theorem is applied to yield very simple proof of characterizations for failure rate orderings of distributions given recently by Capéraà (1988). We also use this composition theorem to give a characterization of two distributions with ordered mean residual life functions.