We study the problem of exact recovery of an unknown multivariate function f observed in the continuous regression model. It is assumed that, in addition to some smoothness constraints, f possesses an additive sparse structure determined by sparsity index β ∈ (0, 1). As a consequence of the additive sparsity assumption, the recovery problem transforms to a variable selection problem. Conditions for exact variable selection are provided and a family of asymptotically minimax variable selection procedures is constructed. The procedures are adaptive with respect to the sparsity index β. Bibliography: 18 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Butucea and Yu. I. Ingster, “Detection of a spare submatrix of a high-dimensional noisy matrix,” http://arxiv.org/abs/1109.0898 (2012).
L. Comminges and A. S. Dalalyan, “Tight conditions for consistency of variable selection in the context of high dimensionality,” http://arxiv.org/abs/1106.4293 (2012).
M. De Groot, Optimal Statistical Decisions, McGraw-Hill Book Company, New York (1970).
M. S. Ermakov, “Minimax detection of a signal in Gaussian white noise,” Teor. Veroyatn. Primen., 35, 667–679 (1991).
G. Gayraud and Yu. I. Ingster, “Detection of sparse variable functions,” http://arxiv.org/abs/1011.6369 (2012).
C. R. Genovese, J. Jin, and L. Wasserman, “Revisiting marginal regression,” http://arxiv.org/abs/0911.4080 (2009).
J. Huang, J. L. Howoritz, and F. Wei, “Variable selection in nonparametric additive models,” Ann. Statist., 2282–2313 (2010).
I. A. Ibragimov and R. Z. Khasminskii, “Some estimation problems on infinite dimensional Gaussian white noise,” in: Festschrift for Lucien Le Cam. Research Papers in Probability and Statistics, Springer-Verlag, New York (1997), pp. 275–296.
Yu. I. Ingster, “Asymptotically minimax hypothesis testing for nonparametric alternatives. I,” Math. Methods Statist., 2, 85–114 (1993).
Yu. I. Ingster, “Asymptotically minimax hypothesis testing for nonparametric. II,” Math. Methods Statist., 2, 171–189 (1993).
Yu. I. Ingster, “Asymptotically minimax hypothesis testing for nonparametric alternatives. III,” Math. Methods Statist., 2, 249–268 (1993).
Yu. I. Ingster and I. A. Suslina, “Nonparametric goodness-of-fit testing under Gaussian models,” Lect. Notes Statist., 169, Springer-Verlag, New York (2003).
Yu. I. Ingster and I. A. Suslina, “On estimation and detection of smooth functions of many variables,” Math. Methods Statist., 14, 299–331 (2005).
Y. Lin, “Tensor product space ANOVA models,” Ann. Statist, 28, 734–755 (2000).
G. Raskutti, M. J. Wainwright, and B. Yu, “Minimax-optimal rates for sparse additive models over kernel classes via convex programming,” http://arxiv.org/abs/1008.3654 (2011).
A. V. Skorokhod, Integration in Hilbert Spaces, Springer-Verlag, Berlin-New York (1974).
V. G. Spokoiny, “Additive hypothesis testing using wavelets,” Ann. Statist., 24, 2477–2498 (1996).
C. J. Stone, “Additive regression and other nonparametric models,” Ann. Statist., 13, 689–705 (1985).
Author information
Authors and Affiliations
Corresponding author
Additional information
Deceased (Yu. Ingster).
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 408, 2012, pp. 214–244.
Rights and permissions
About this article
Cite this article
Ingster, Y., Stepanova, N. Adaptive Variable Selection in Nonparametric Sparse Regression. J Math Sci 199, 184–201 (2014). https://doi.org/10.1007/s10958-014-1846-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1846-7