Order-preserving property of maximum likelihood estimator

Various properties of the maximum likelihood estimator have been studied. However, the order preserving property of MLE has not been studied in the literature. This is precisely the subject matter of this paper. General sufficient conditions are derived under which the MLE possesses the property of preserving the usual stochastic order. In particular, sufficient conditions are obtained for exponential family, location family, scale family, and failure rate models. Examples are finally presented to illustrate the results obtained in this paper.