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On adaptive nonparametric inference

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One of the intriguing problems in the paradigm of adaptive nonparametric function estimation as developed in the last two decades by Donoho, Johnstone, Lepski, Birge, Massart, Kerkyacharian, Picard, Tsybakov, Spokoiny, Cai, Low and many others, is what one could call the ‘hiatus’ between estimation and inference, or, to be more precise, between the existence of adaptive risk bounds and the non-existence of adaptive confidence statements. Considering the fundamental problem of the existence of adaptive confidence bands, we show that this fact is deeply connected to certain nonparametric hypothesis testing problems studied by the Russian school (Ingster and others), and we use this connection to derive necessary and sufficient conditions for the existence of adaptive confidence statements, thus characterising the exact boundaries of when adaptive inference is possible and when not.

This talk is part of the Statistics series.

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