Multivariate wavelet thresholding: a remedy against the curse of dimensionality?
- Neumann, Michael H.
2010 Mathematics Subject Classification
- 62G07 62G20
- Nonparametric curve estimation, multivariate wavelet estimators, nonlinear thresholding, curse of dimensionality, anisotropic wavelet basis, anisotropic smoothness classes, smoothness classes with dominating mixed derivatives, optimal rate of convergence
It is well-known that multivariate curve estimation suffers from the "curse of dimensionality". However, reasonable estimators are possible, even in several dimensions, under appropriate restrictions on the complexity of the curve. In the present paper we explore how much appropriate wavelet estimators can exploit typical restrictions on the curve, which require a local adaptation to different degrees of smoothness in the different directions. It turns out that the application of a anisotropic multivariate basis, which has in contrast to the conventional multivariate resolution scheme a multidimensional scale parameter, is essential. Some simulations indicate the possible gains by this new method over thresholded estimators based on the multiresolution basis with a one-dimensional scale index.