Adaptive testing on a regression function at a point

TL;DR

This paper develops an adaptive testing method for regression functions at a point, achieving near-optimal rates under shape constraints, with applications to regression discontinuity, boundary values, and multiple testing.

Contribution

It introduces a new adaptive test that attains the minimax rate over various Hölder classes, incorporating shape restrictions and demonstrating the necessity of a log-log factor.

Findings

Achieves near-minimax optimal rates for adaptive testing.

Applies to regression discontinuity and boundary value problems.

Handles multiple testing scenarios with shape constraints.

Abstract

We consider the problem of inference on a regression function at a point when the entire function satisfies a sign or shape restriction under the null. We propose a test that achieves the optimal minimax rate adaptively over a range of H\"{o}lder classes, up to a $\log\log n$ term, which we show to be necessary for adaptation. We apply the results to adaptive one-sided tests for the regression discontinuity parameter under a monotonicity restriction, the value of a monotone regression function at the boundary and the proportion of true null hypotheses in a multiple testing problem.