Statistica Sinica 13(2003), 881-902

RATES OF CONVERGENCE AND ADAPTATION OVER

BESOV SPACES UNDER POINTWISE RISK

T. Tony Cai

University of Pennsylvania

Abstract: Function estimation over the 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.

Key words and phrases: Adaptability, adaptive estimation, Besov spaces, constrained risk inequality, minimax estimation, nonparametric functional estimation, oracle inequality, rate of convergence, wavelets, white noise model.