The Annals of Statistics

Global uniform risk bounds for wavelet deconvolution estimators

Karim Lounici and Richard Nickl

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Abstract

We consider the statistical deconvolution problem where one observes n replications from the model Y = X + ϵ, where X is the unobserved random signal of interest and ϵ is an independent random error with distribution φ. Under weak assumptions on the decay of the Fourier transform of φ, we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators fn for the density f of X, where f : ℝ → ℝ is assumed to be bounded. We then derive lower bounds for the minimax sup-norm risk over Besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds. We further show that linear estimators adapt to the unknown smoothness of f if the Fourier transform of φ decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if φ decays polynomially. We also analyze the case where f is a “supersmooth”/analytic density. We finally show how our results and recent techniques from Rademacher processes can be applied to construct global confidence bands for the density f.

Article information

Source
Ann. Statist. Volume 39, Number 1 (2011), 201-231.

Dates
First available in Project Euclid: 3 December 2010

Permanent link to this document
http://projecteuclid.org/euclid.aos/1291388373

Digital Object Identifier
doi:10.1214/10-AOS836

Mathematical Reviews number (MathSciNet)
MR2797844

Zentralblatt MATH identifier
05874493

Subjects
Primary: 62G07: Density estimation
Secondary: 62G15: Tolerance and confidence regions

Keywords
Band-limited wavelets sup-norm loss Vapnik–Chervonenkis class confidence band Rademacher process

Citation

Lounici, Karim; Nickl, Richard. Global uniform risk bounds for wavelet deconvolution estimators. Ann. Statist. 39 (2011), no. 1, 201--231. doi:10.1214/10-AOS836. http://projecteuclid.org/euclid.aos/1291388373.


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References

  • [1] Bartlett, P., Boucheron, S. and Lugosi, G. (2002). Model selection and error estimation. Machine Learning 48 85–113.
  • [2] Bergh, J. and Löfström, J. (1976). Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften 223. Springer, Berlin.
  • [3] Bissantz, N., Dümbgen, L., Holzmann, H. and Munk, A. (2007). Non-parametric confidence bands in deconvolution density estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 483–506.
  • [4] Bissantz, N. and Holzmann, H. (2008). Statistical inference for inverse problems. Inverse Problems 24 034009.
  • [5] Bourdaud, G., Lanza de Cristoforis, M. and Sickel, W. (2006). Superposition operators and functions of bounded p-variation. Rev. Mat. Iberoamericana 22 455–487.
  • [6] Bousquet, O. (2003). Concentration inequalities for sub-additive functions using the entropy method. In Stochastic Inequalities and Applications. Progress in Probability 56 213–247. Birkhäuser, Basel.
  • [7] Butucea, C. and Tsybakov, A. B. (2008a). Sharp optimality in density deconvolution with dominating bias. I. Theory Probab. Appl. 52 24–39.
  • [8] Butucea, C. and Tsybakov, A. B. (2008b). Sharp optimality in density deconvolution with dominating bias. II. Theory Probab. Appl. 52 237–249.
  • [9] Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.
  • [10] Cavalier, L. (2008). Nonparametric statistical inverse problems. Inverse Problems 24 034004.
  • [11] Delaigle, A. and Gijbels, I. (2004). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249–267.
  • [12] Diggle, P. J. and Hall, P. (1993). A Fourier approach to nonparametric deconvolution of a density estimate. J. Roy. Statist. Soc. Ser. B 55 523–531.
  • [13] Einmahl, U. and Mason, D. M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 1–37.
  • [14] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
  • [15] Fan, J. (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 600–610.
  • [16] Giné, E. and Guillou, A. (2002). Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré Probab. Statist. 38 907–921.
  • [17] Giné, E., Koltchinskii, V. and Sakhanenko, L. (2004). Kernel density estimators: Convergence in distribution for weighted sup-norms. Probab. Theory Related Fields 130 167–198.
  • [18] Giné, E., Latała, R. and Zinn, J. (2000). Exponential and moment inequalities for U-statistics. In High Dimensional Probability, II. Progress in Probability 47 13–38. Birkhäuser, Boston, MA.
  • [19] Giné, E. and Nickl, R. (2009). Uniform limit theorems for wavelet density estimators. Ann. Probab. 37 1605–1646.
  • [20] Giné, E. and Nickl, R. (2010). Confidence bands in density estimation. Ann. Statist. 38 1122–1170.
  • [21] Giné, E. and Nickl, R. (2010). Adaptive estimation of the distribution function and its density by wavelet and spline projections. Bernoulli. To appear.
  • [22] Goldenshluger, A. (1999). On pointwise adaptive nonparametric deconvolution. Bernoulli 5 907–925.
  • [23] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statist. 129. Springer, New York.
  • [24] Hesse, C. H. and Meister, A. (2004). Optimal iterative density deconvolution. J. Nonparametr. Statist. 16 879–900.
  • [25] Johnstone, I. M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 547–573.
  • [26] Johnstone, I. M. and Raimondo, M. (2004). Periodic boxcar deconvolution and Diophantine approximation. Ann. Statist. 32 1781–1804.
  • [27] Klein, T. and Rio, E. (2005). Concentration around the mean for maxima of empirical processes. Ann. Probab. 33 1060–1077.
  • [28] Koltchinskii, V. (2001). Rademacher penalties and structural risk minimization. IEEE Trans. Inform. Theory 47 1902–1914.
  • [29] Koltchinskii, V. (2006). Local Rademacher complexities and oracle inequalities in risk minimization. Ann. Statist. 34 2593–2656.
  • [30] Meister, A. (2008). Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions. Inverse Problems 24 015003.
  • [31] Meister, A. (2009). Deconvolution Problems in Nonparametric Statistics. Springer, Berlin.
  • [32] Meyer, Y. (1992). Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37. Cambridge Univ. Press, Cambridge.
  • [33] Nickl, R. and Pötscher, B. M. (2007). Bracketing metric entropy rates and empirical central limit theorems for function classes of Besov- and Sobolev-type. J. Theoret. Probab. 20 177–199.
  • [34] Nolan, D. and Pollard, D. (1987). U-processes: Rates of convergence. Ann. Statist. 15 780–799.
  • [35] Pensky, M. and Sapatinas, T. (2009). Functional deconvolution in a periodic setting: Uniform case. Ann. Statist. 37 73–104.
  • [36] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033–2053.
  • [37] Stefanski, L. A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229–235.
  • [38] Stefanski, L. A. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169–184.
  • [39] Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505–563.
  • [40] Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, Berlin.