## Electronic Journal of Statistics

### Honest and adaptive confidence sets in $L_{p}$

Alexandra Carpentier

#### Abstract

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.

#### Article information

Source
Electron. J. Statist., Volume 7 (2013), 2875-2923.

Dates
First available in Project Euclid: 2 December 2013

https://projecteuclid.org/euclid.ejs/1385995294

Digital Object Identifier
doi:10.1214/13-EJS867

Mathematical Reviews number (MathSciNet)
MR3148371

Zentralblatt MATH identifier
1280.62055

Subjects
Primary: 62G15: Tolerance and confidence regions
Secondary: 62G10: Hypothesis testing

#### Citation

Carpentier, Alexandra. Honest and adaptive confidence sets in $L_{p}$. Electron. J. Statist. 7 (2013), 2875--2923. doi:10.1214/13-EJS867. https://projecteuclid.org/euclid.ejs/1385995294

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