LAUSR

Shape-constrained inference usually refers to nonparametric function estimation and uncertainty quantification under qualitative shape restrictions such as monotonicity, convexity, log-concavity and so on. One of the earliest contributions to the field was by Grenander (1956). Motivated by the theory of mortality measurement, he studied the nonparametric maximum likelihood estimator of a decreasing density function on the nonnegative half-line. A great attraction of this estimator is that, unlike other nonparametric density estimators such as histograms or kernel density estimators, there are no tuning parameters (e.g., bandwidths) to choose. Over subsequent years, this idea has been extended and developed in many different directions. On the applied side, there has been a gradual realisation that nonparametric shape constraints are very natural to impose in many situations. For instance, monotonicity of a regression function arises in many contexts such as genetics (Luss, Rosset and Shahar, 2012), medicine (Schell and Singh, 1997) and dose-response modelling (Lin et al., 2012). Shape-constrained procedures are also commonly used in economics (Matzkin, 1991, Varian, 1984) and survival analysis, for instance in the interval-censoring problem and hazard function estimation; see the recent book by Groeneboom and Jongbloed (2014). Many other applications, and further developments, including the computational aspects of these shape-constrained estimators, are nicely summarised in the books by Barlow et al. (1972), Robertson, Wright and Dykstra (1988) and Groeneboom and Wellner (1992). On the theoretical side, it has been known since the work of Prakasa Rao (1969) that the Grenander