Statistics Papers

Document Type

Journal Article

Date of this Version

2013

Publication Source

The Annals of Statistics

Volume

41

Issue

2

Start Page

722

Last Page

750

DOI

10.1214/12-AOS1068

Abstract

Adaptive confidence intervals for regression functions are constructed under shape constraints of monotonicity and convexity. A natural benchmark is established for the minimum expected length of confidence intervals at a given function in terms of an analytic quantity, the local modulus of continuity. This bound depends not only on the function but also the assumed function class. These benchmarks show that the constructed confidence intervals have near minimum expected length for each individual function, while maintaining a given coverage probability for functions within the class. Such adaptivity is much stronger than adaptive minimaxity over a collection of large parameter spaces.

Keywords

adaptation, confidence interval, convex function, coverage probability, expected length, minimax estimation, modulus of continuity, monotone function, nonparametric regression, shape constraint, white noise model

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Date Posted: 27 November 2017

This document has been peer reviewed.