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

Cai, T. Tony

Stat. Sin. 13, No. 3, 881-902 (2003).

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

Keywords:

adaptability; adaptive estimation; Besov spaces; constrained risk inequality; minimax estimation; functional estimation; oracle inequality; rate of convergence; wavelets; white noise model