Estimation in shape-constrained statistical models dates back at least as far as Hildreth [(1954); J. Amer. Statist. Assoc., 49, 598–619], who considered the maximum likelihood estimation of production functions under the natural assumption of non-increasing returns (which implies that the production function is concave and nondecreasing). Around the same time, Grenander [(1956); Skand. Aktuarietidskr., 39, 125–153], motivated by the theory of mortality measurement, studied the nonparametric maximum likelihood estimator of a decreasing density function on the positive half line. Over subsequent years, these ideas have been extended and developed in many different directions.
Shape-constrained methods may be applied to regression function estimation as well as density estimation, and allow the user to implement vague and qualitative assumptions about functional forms, without having to specify parametric models. This is extremely useful in the common situation where the only valid assumptions involve shape (and smoothness). Being nonparametric, the methods are more robust than standard parametric approaches. Further, although these methods deal with infinite- dimensional models (e.g., functions), estimation can still be carried out using the method of maximum likelihood (probably the main technique for estimation of parameters in a statistical model).