On Minimax Wavelet Estimators

Abstract

In the paper minimax rates of convergence for wavelet estimators are studied. The estimators are based on the shrinkage of empirical coefficients βjkof wavelet decomposition of unknown function with thresholds λj. These thresholds depend on the regularity of the function to be estimated. In the problem of density estimation and nonparametric regression we establish upper rates of convergence over a large range of functional classes and global error measures. The constructed estimate is minimax (up to constant) for all Lπerror measures, 0 < π ≤ ∞ simultaneously.