Adaptive Regression on the Real Line in Classes of Smooth Functions
DOI:
https://doi.org/10.17713/ajs.v32i1&2.452Abstract
Adaptive pointwise estimation of an unknown regression function f(x), x ? R corrupted by additive Gaussian noise is considered in the equidistant design setting. The function f is assumed to belong to the class A(?) of functions whose Fourier transform are rapidly decreasing in the weighted L2-sense. The rate of decrease is described by a weight function that depends on the vector of parameters ? which, in the adaptive setting, is typically unknown. For any of the classes A(?) , ? fixed, we describe minimax estimators up to a constant as the bin-width goes to zero. Conditions under which an adaptive study is suitable are presented and a notion of adaptive asymptotic optimality is introduced based on distinguishing, among all possible functional scales, between the so-called non-parametric (NP) and pseudo-parametric (PP) scales. We propose adaptive estimators which ‘tune up’ point-wisely to the unknown smoothness of f. We prove them to be asymptotically adaptively minimax for large collections of NP functional scales, subject to being rate efficient for any of the PP functional
scales.
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