Asymptotically minimax non–parametric function estimation with positivity constraints I

Abstract

The spatially inhomogeneous smoothness of the non-parametric density or regression-function to be estimated by non-parametric methods is often modelled by Besov- and Triebel-type smoothness constraints. For such problems, Donoho and Johnstone [D.L. Donoho and I.M. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1998), pp. 879–921.], Delyon and Juditsky [B. Delyon and A. Juditsky, On minimax wavelet estimators, Appl. Comput. Harmon. Anal. 3 (1996), pp. 215–228.] studied minimax rates of convergence for wavelet estimators with thresholding, while Lepski et al. [O.V. Lepski, E. Mammen, and V.G. Spokoiny, Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimators with variable bandwidth selectors, Ann. Stat. 25 (1997), pp. 929–947.] proposed a variable bandwidth selection for kernel estimators that achieved optimal rates over Besov classes. However, a second challenge in many real applications of non-parametric curve estimation is that the function must be positive. Here, we show how to construct estimators under positivity constraints that satisfy these constraints and also achieve minimax rates over the appropriate smoothness class.

Keywords:

• wavelet estimator
• kernel estimator
• Besov space
• Triebel–Lizorkin space
• Wiener–Young bounded variation

AMS Subject Classification :

• Primary: 62G07
• Secondary: 62G05
• 41A65
• 46B40
• 46B70
• 46E30

Acknowledgements

We are thankful to Dr. Spiridon Penev (University of New South Wales, Sydney, Australia) for making the manuscripts 2 3 available to us while they were still in preprint form. We thank the referees for some very useful remarks which helped us improve the paper. We also wish to thank both the Editor and the referees for their patience and their understanding of the circumstances which led to the delay of the revision of this paper of several years. This research has been supported by the Natural Sciences and Engineering Research Council of Canada and, in its final stage, by the 2007 Annual Research Grant of the R&D Group for Mathematical Modelling, Numerical Simulation and Computer Visualization at Narvik University College, Norway.