On the Kuhn-Tucker equivalence theorem and its applications to isotonic regression

Prior information regarding a statistical model frequently constrains the shape of the parameter set and can often be quantified by placing inequality constraints on the parameters. For example, a regression function may be nondecreasing or convex or both; or the treatment response may stochastically dominate the control. The order restricted statistical inference has been well developed since the 1950's. The isotonic regression solves many restricted maximum likelihood estimation problems. And the theory of duality (cf. Barlow and Brunk (1972)) has provided insights into new problems. Both the isotonic regression and Fenchel duality play the important roles in order restricted statistical inference. -- Kuhn and Tucker (1951) proposed a necessary and sufficient condition for the solution to an inequality constrained maximization problem. Since then, the Kuhn-Tucker equivalence theorem has been extensively applied to many fields such as optimization theory, engineering, the economy and so on. -- In this paper, we focus on the applications of the Kuhn-Tucker equivalence theorem to order restricted estimation. This equivalence theorem provides a completely different approach to prove many important results such as the generalized isotonic regression problem due to Barlow and Brunk (1972), I-projection problems due to Dykstra (1985) and so on. We provide some insights into its extensive applications to ordered statistical inference. We expect that the kuhn-Tucker equivalence theorem will become a powerful tool in this field. -- Key words: Convex cone; Isotonic regression; Linearity space; Partial ordering; Polyhedral convex sets; Simple tree ordering; Van Eeden's algorithm