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Honest and adaptive confidence sets in $$L_p$$. (English) Zbl 1280.62055
Summary: We consider the problem of constructing honest and adaptive confidence sets in $$L_p$$-loss (with $$p\geq 1$$ and $$p < \infty$$) over sets of Sobolev-type classes, in the setting of nonparametric 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 $$p \geq 2$$, 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.

##### MSC:
 62G15 Nonparametric tolerance and confidence regions 62G08 Nonparametric regression and quantile regression 62G10 Nonparametric hypothesis testing
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