The Annals of Statistics

Adaptive confidence intervals for regression functions under shape constraints

T. Tony Cai, Mark G. Low, and Yin Xia

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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.

Article information

Source
Ann. Statist., Volume 41, Number 2 (2013), 722-750.

Dates
First available in Project Euclid: 8 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1368018171

Digital Object Identifier
doi:10.1214/12-AOS1068

Mathematical Reviews number (MathSciNet)
MR3099119

Zentralblatt MATH identifier
1267.62066

Subjects
Primary: 62G99: None of the above, but in this section
Secondary: 62F12: Asymptotic properties of estimators 62F35: Robustness and adaptive procedures 62M99: None of the above, but in this section

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

Citation

Cai, T. Tony; Low, Mark G.; Xia, Yin. Adaptive confidence intervals for regression functions under shape constraints. Ann. Statist. 41 (2013), no. 2, 722--750. doi:10.1214/12-AOS1068. https://projecteuclid.org/euclid.aos/1368018171


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