Inference for Local Parameters in Convexity Constrained Models

In this article, we develop automated inference methods for “local” parameters in a collection of convexity constrained models based on the natural constrained tuning-free estimators. A canonical example is given by the univariate convex regression model, in which automated inference is drawn for the function value, the function derivative at a fixed interior point, and the anti-mode of the convex regression function, based on the widely used tuning-free, piecewise linear convex least squares estimator (LSE). The key to our inference proposal in this model is a pivotal joint limit distribution theory for the LS estimates of the local parameters, normalized appropriately by the length of certain data-driven linear piece of the convex LSE. Such a pivotal limiting distribution instantly gives rise to confidence intervals for these local parameters, whose construction requires almost no more effort than computing the convex LSE itself. This inference method in the convex regression model is a special case of a general inference machinery that covers a number of convexity constrained models in which a limit distribution theory is available for model-specific estimators. Concrete models include: (i) log-concave density estimation, (ii) s-concave density estimation, (iii) convex nonincreasing density estimation, (iv) concave bathtub-shaped hazard function estimation, and (v) concave distribution function estimation from corrupted data. The proposed confidence intervals for all these models are proved to have asymptotically exact coverage and oracle length, and require no further information than the estimator itself. We provide extensive simulation evidence that validates our theoretical results. Real data applications and comparisons with competing methods are given to illustrate the usefulness of our inference proposals. Supplementary materials for this article are available online.