Translator Disclaimer
2013 Honest and adaptive confidence sets in $L_{p}$
Alexandra Carpentier
Electron. J. Statist. 7: 2875-2923 (2013). DOI: 10.1214/13-EJS867


We consider the problem of constructing honest and adaptive confidence sets in $L_{p}$-loss (with $p\geq1$ and $p<\infty$) over sets of Sobolev-type classes, in the setting of non-parametric Gaussian regression. The objective is to adapt the diameter of the confidence sets with respect to the smoothness degree of the underlying function, while ensuring that the true function lies in the confidence interval with high probability. When $p\geq2$, we identify two main regimes, (i) one where adaptation is possible without any restrictions on the model, and (ii) one where critical regions have to be removed. We also prove by a matching lower bound that the size of the regions that we remove can not be chosen significantly smaller. These regimes are shown to depend in a qualitative way on the index $p$, and a continuous transition from $p=2$ to $p=\infty$ is exhibited.


Download Citation

Alexandra Carpentier. "Honest and adaptive confidence sets in $L_{p}$." Electron. J. Statist. 7 2875 - 2923, 2013.


Published: 2013
First available in Project Euclid: 2 December 2013

zbMATH: 1280.62055
MathSciNet: MR3148371
Digital Object Identifier: 10.1214/13-EJS867

Primary: 62G15
Secondary: 62G10

Rights: Copyright © 2013 The Institute of Mathematical Statistics and the Bernoulli Society


Back to Top