Adaptive confidence sets in L^{2}

Details Adaptive confidence sets in L^{2} Adaptive confidence sets in L^{2}

2011-11-23

arxiv.org/abs/1111.5568v1

Mathematics

Statistics Theory

Scientific paper

The problem of constructing nonparametric confidence sets that are adaptive in L^{2}-loss over a continuous scale of Sobolev classes is considered. Adaptation holds, where possible, with respect to both the radius of the Sobolev ball and its smoothness degree, and over maximal parameter spaces for which adaptation is possible. Two key regimes of parameter constellations are identified: one where full adaptation is possible, and one where adaptation requires critical regions be removed. The phase transition between these regimes is analysed separately. Key ideas needed to derive these results include a general nonparametric minimax test for infinite-dimensional null- and alternative hypotheses, and new lower bound techniques for L^{2}-adaptive confidence sets.

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