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April 2013 Adaptive confidence intervals for regression functions under shape constraints
T. Tony Cai, Mark G. Low, Yin Xia
Ann. Statist. 41(2): 722-750 (April 2013). 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.

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T. Tony Cai. Mark G. Low. Yin Xia. "Adaptive confidence intervals for regression functions under shape constraints." Ann. Statist. 41 (2) 722 - 750, April 2013. https://doi.org/10.1214/12-AOS1068

Information

Published: April 2013
First available in Project Euclid: 8 May 2013

zbMATH: 1267.62066
MathSciNet: MR3099119
Digital Object Identifier: 10.1214/12-AOS1068

Subjects:
Primary: 62G99
Secondary: 62F12 , 62F35 , 62M99

Keywords: Adaptation , Confidence interval , convex function , coverage probability , expected length , minimax estimation , modulus of continuity , monotone function , Nonparametric regression , shape constraint , White noise model

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • April 2013
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