Wavelets and Optimal Nonlinear Function Estimates

We consider the problem of estimating a smooth function from noisy, sampled data. We use orthonormal bases of compactly supported wavelets to constriuct nonlinear function estimates which can significantly outperform evey linear method (kernel, smoothing spline, sieve, ...). Our estimates are simple nonlinear functions of the empirical wavelet coefficients and are asymptotically minimax over certain Besov smoothness classes. Our estimates possess the interpretation of local adaptiveness: they reconstruct using a kernel which may vary in shape and bandwidth from point to point, depending on the data. Modifications of our estimates based on simple threshold nonlinearities are near minimax and have interesting interpretations as smoothness-penalized least squares estimates or as adaptive depleted-basis spline fits.