Adaptive confidence sets in shape restricted regression

Abstract

A simple construction of adaptive confidence sets is proposed in isotonic, convex and unimodal regression. In univariate isotonic regression, the proposed confidence set enjoys uniform coverage over all non-decreasing regression functions. Furthermore, the diameter of the proposed confidence set automatically adapts to the unknown number of pieces of the true parameter, in the sense that the diameter is bounded from above by the minimax risk over the class of $k$-piecewise constant functions. The diameter of the confidence set is a simple increasing function of the number of jumps of the isotonic least-squares estimate.

A similar construction is proposed in convex regression where the true regression function is convex and piecewise affine. Here, the confidence set enjoys uniform coverage and its diameter automatically adapts to the number of affine pieces of the true regression function. The diameter of the confidence set is an increasing function of the number of affine pieces of the convex least-squares estimate.

We explain how to extend this technique to a non-convex set by proposing a similar adaptive confidence set in unimodal regression. The confidence set automatically adapts to the number of jumps of the true unimodal regression function and its diameter is an increasing function of the number of jumps of the unimodal least-squares estimate.