Summary: The author and I. M. Johnstone [Biometrika 81, No. 3, 425–455 (1994; Zbl 0815.62019)] proposed a method for reconstructing an unknown function on from noisy data , , , where the are independent and identically distributed standard Gaussian random variables. The reconstruction is defined in the wavelet domain by translating all the empirical wavelet coefficients of toward 0 by an amount . We prove two results about this type of estimator.
[Smooth]: With high probability is at least as smooth as , in any of a wide variety of smoothness measures.
[Adapt]: The estimator comes nearly as close in mean square to as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes.
These two properties are unprecedented in several ways. Our proof of these results develops new facts about abstract statistical inference and its connection with an optimal recovery model.