Properties of Likelihood Inference for Order Restricted Models

The most common mode of inference for order restricted models is likelihood inference. See T. Robertson, F. T. Wright, and R. L. Dykstra (1988, "Order Restricted Statistical Inference," Wiley, New York) for an excellent treatment of inference in such models. In this paper we demonstrate that maximum likelihood estimation and likelihood ratio testing are prone to behavior that is somewhat unintuitive and unappealing to our sensibilities for many order restricted models. From a practical standpoint this behavior sometimes is cause enough for us to seek alternative procedures. For restrictions to a simple order cone, likelihood inference is satisfactory. However, if, for example, the restrictions are of the tree order type, umbrella order, star-shaped order, or stochastic order, then likelihood methods seem to have some shortcomings. General results will identify when likelihood methods are potentially wanting.