Structural adaptation in the single-index model

Details Structural adaptation in the single-index model Structural adaptation in the single-index model

2011-11-15

arxiv.org/abs/1111.3563v2

Mathematics

Statistics Theory

23 pages. This paper has been withdrawn by the second author due to its incompleteness for future improvement

Scientific paper

We study the problem of adaptive estimation of a multivariate function under the single-index constrains when both the link function and index vector are unknown. We propose a novel estimation procedure adapting simultaneously to the unknown index vector and the smoothness of the link function by selection from a family of specific kernel estimators. This procedure allows to establish a pointwise oracle inequality which can be used to judge the quality of estimation under the L_r-norm losses. The obtained oracle inequality allows adaptation with respect to a scale of Nikol'skii (Besov) classes B_{p, \infty}^s, where s is the unknown regularity of link function and p is the index of L_p-norm involved in the class definition. This paper therefore covers non-treated in the literature on structural adaptation and, correspondingly by the recent Goldenshluger-Lepski procedure, case of r > p. We also provide the lower bounds for the considered class of functions possessing the single index structure.

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