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