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Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection
Keywords: Anisotropic Besov spaces; Multivariate estimation
Thu, 09:25--09:50
  • Akakpo, Nathalie (Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, France)

When estimating a multivariate function, it seems natural to consider that its smoothness is likely to vary either spatially, or with the direction, or both. We will refer to the first feature as (spatial) inhomogeneity and to the second one as anisotropy. In particular, we can thus take into account multivariate functions that may in fact be constant in some of the variables. Yet, statistical procedures that adapt both to possible inhomogeneity and anisotropy remain rather scarce. Results of this type rely as much on Statistics as on Approximation Theory.

So we propose an approximation result devoted to piecewise polynomial approximation based on partitions into dyadic rectangles, inspired from DeVore and Yu [1], over possibly inhomogeneous and anisotropic smoothness classes that contain for instance the more traditional Besov classes. Besides, we take into account a possible restriction on the minimal size of the dyadic rectangles, which may arise in statistical applications. For estimating a multivariate function –in the density estimation or regression framework, for instance–, we then introduce a selection procedure that chooses from the data the best partition into dyadic rectangles and the best piecewise polynomial built on that partition thanks to a penalized least-squares type criterion. The degree of the polynomial may vary from one rectangle to another, and also according to the coordinate directions. Such a procedure is able to reach the optimal estimation rate although the exact degree of smoothness is unknown. This results not only from the good approximation properties of dyadic piecewise polynomials, but also from the moderate number of dyadic partitions of the same size. Moreover, our statistical procedure can be implemented with a computational complexity only linear in the sample size.

[1] R. A. DeVore and X. M. Yu. Degree of adaptive approximation. Math. Comp., 55(192):625–635, 1990.