Properties of Likelihood Inference for Order Restricted Models

• Published in: Journal of multivariate analysis Vol. 72; no. 1; pp. 50 - 77
• Main Authors: Cohen, Arthur, Kemperman, J.H.B, Sackrowitz, Harold B
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.01.2000; Elsevier
• Series: Journal of Multivariate Analysis
• Subjects: cone order; cone order monotonicity; convex cones; convex cones, projection, preservation, reversal, cone order, cone order monotonicity, simple order, tree order, star-shaped order, umbrella order, matrix order, stochastic order, maximum likelihood estimators, likelihood ratio tests; Exact sciences and technology; likelihood ratio tests; Mathematics; matrix order; maximum likelihood estimators; Multivariate analysis; preservation; Probability and statistics; projection; reversal; Sciences and techniques of general use; simple order; star-shaped order; Statistics; stochastic order; tree order; umbrella order; convex cones, projection, preservation, reversal, cone order, cone order monotonicity, simple order, tree order, star-shaped order, umbrella order, matrix order, stochastic order, maximum likelihood estimators, likelihood ratio tests; Stochastic analysis; Cone monotonicity order; Preservation; Likelihood ratio test; Stochastic order; Projection; Maximum likelihood; Convex cone
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1006/jmva.1999.1847

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.