Multivariate Wavelet Thresholding: a Remedy against the Curse of Dimensionality?

I thank V. Spokoiny for helpful comments on this paper. The research was carried out within the Sonderforschungsbereich 373 at Humboldt University Berlin and was printed using funds made available by the Deutsche Forschungsgemeinschaft. Abstract. It is well-known that multivariate curve estimation suuers 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 diierent degrees of smoothness in the diierent directions. It turns out that the application of a anisotropic multivariate basis, which has in contrast to the conventional mul-tivariate resolution scheme a multidimensional scale parameter, is essential. Some simulations indicate the possible gains by this new method over thresholded esti-mators based on the multiresolution basis with a one-dimensional scale index. 1. Introduction Multivariate curve estimation is often considered with some scepticism, because it is associated with the term of the \curse of dimensionality". This notion reeects the fact that nonparametric statistical methods lose much of their power if the dimension d is large. In the presence of r bounded derivatives, the usual optimal rate of convergence in regression or density estimation is n ?2r=(2r+d) , where n denotes the number of observations. To get the same rate as in the one-dimensional case, one has to assume a smoothness of order rd rather than r. This phenomenon can also be explained by the sparsity of data in high dimensions. If we have a uniformly distributed sample over the hypercube ?1; 1] d , then we will nd only a fraction of about 2 ?d of the data in the hypercube 0; 1] d. Nevertheless, there is sometimes some hope for a successful statistical analysis in higher dimensions. Often the true complexity of a multivariate curve is much lower than it could be expected from a statement that the curve is a member of a certain Sobolev class W r p (R d) with degree of smoothness r. Scott (1992, Chapter 7) claims: