Nonparametric regression estimation based on spatially inhomogeneous data: Minimax global convergence rates and adaptivity
Date
2014ISSN
12928100Source
ESAIM  Probability and StatisticsVolume
18Pages
141Google Scholar check
Keyword(s):
Metadata
Show full item recordAbstract
We consider the nonparametric regression estimation problem of recovering an unknown response function f on the basis of spatially inhomogeneous data when the design points follow a known density g with a finite number of wellseparated zeros. In particular, we consider two different cases: when g has zeros of a polynomial order and when g has zeros of an exponential order. These two cases correspond to moderate and severe data losses, respectively. We obtain asymptotic (as the sample size increases) minimax lower bounds for the L2risk when f is assumed to belong to a Besov ball, and construct adaptive wavelet thresholding estimators of f that are asymptotically optimal (in the minimax sense) or nearoptimal within a logarithmic factor (in the case of a zero of a polynomial order), over a wide range of Besov balls. The spatially inhomogeneous illposed problem that we investigate is inherently more difficult than spatially homogeneous illposed problems like, e.g., deconvolution. In particular, due to spatial irregularity, assessment of asymptotic minimax global convergence rates is a much harder task than the derivation of asymptotic minimax local convergence rates studied recently in the literature. Furthermore, the resulting estimators exhibit very different behavior and asymptotic minimax global convergence rates in comparison with the solution of spatially homogeneous illposed problems. For example, unlike in the deconvolution problem, the asymptotic minimax global convergence rates are greatly influenced not only by the extent of data loss but also by the degree of spatial homogeneity of f. Specifically, even if 1/g is nonintegrable, one can recover f as well as in the case of an equispaced design (in terms of asymptotic minimax global convergence rates) when it is homogeneous enough since the estimator is "borrowing strength" in the areas where f is adequately sampled. © EDP Sciences, SMAI 2013.
Collections
Cite as
Related items
Showing items related by title, author, creator and subject.

Conference Object
Robust estimation with applications to phase and envelope estimation in frequency selective wireless fading channels
Socratous, Y.; Charalambous, Charalambos D.; Georghiades, C. N. (2008)This paper derives robust niiniinax estimators for a class of uncertain models. The uncertainty is described by a relative entropy constraint between the unknown joint distribution and a fixed nominal joint distribution. ...

Conference Object
Distributed network size estimation and average degree estimation and control in networks isomorphic to directed graphs
Shames, I.; Charalambous, T.; Hadjicostis, Christoforos N.; Johansson, M. (2012)Many properties of interest in graph structures are based on the nodes' average degree (i.e., the average number of edges incident to/from each node). In this work, we present asynchronous distributed algorithms, based on ...

Conference Object
LeastSquare estimation for nonlinear systems with applications to phase and envelope estimation in wireless fading channels
Socratous, Y.; Charalambous, Charalambos D.; Georghiades, C. N. (2008)This paper addresses the problem of nonlinear LeastSquare estimation. A new approach is presented which employees a change of probability measure technique to derive recursive equations for conditional means of nonlinear ...