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Rates of convergence and adaptation over Besov spaces under pointwise risk. (English) Zbl 1046.62046

Summary: Function estimation over Besov spaces under pointwise \(\ell^r(1\leq r<\infty)\) risks is considered. Minimax rates of convergence are derived using a constrained risk inequality and wavelets. Adaptation under pointwise risks is also considered. Sharp lower bounds on the cost of adaptation are obtained and are shown to be attainable by a wavelet estimator. The results demonstrate important differences between the minimax properties under pointwise and global risk measures. The minimax rates and adaptation for estimating derivatives under pointwise risks are also presented. A general \(\ell^r\)-risk oracle inequality is developed for the proofs of the main results.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
62C20 Minimax procedures in statistical decision theory
62M09 Non-Markovian processes: estimation
46N30 Applications of functional analysis in probability theory and statistics
62M05 Markov processes: estimation; hidden Markov models
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