The Annals of Statistics
- Ann. Statist.
- Volume 41, Number 2 (2013), 722-750.
Adaptive confidence intervals for regression functions under shape constraints
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.
Ann. Statist., Volume 41, Number 2 (2013), 722-750.
First available in Project Euclid: 8 May 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
Adaptation confidence interval convex function coverage probability expected length minimax estimation modulus of continuity monotone function nonparametric regression shape constraint white noise model
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