Shape-Constrained Statistical Inference

Abstract

Statistical models defined by shape constraints are a valuable alternative to parametric models or nonparametric models defined in terms of quantitative smoothness constraints. While the latter two classes of models are typically difficult to justify a priori, many applications involve natural shape constraints, for instance, monotonicity of a density or regression function. We review some of the history of this subject and recent developments, with special emphasis on algorithmic aspects, adaptivity, honest confidence bands for shape-constrained curves, and distributional regression, i.e., inference about the conditional distribution of a real-valued response given certain covariates.