Electronic Journal of Statistics

A uniform central limit theorem and efficiency for deconvolution estimators

Jakob Söhl and Mathias Trabs

Full-text: Open access

Abstract

We estimate linear functionals in the classical deconvolution problem by kernel estimators. We obtain a uniform central limit theorem with $\sqrt{n}$–rate on the assumption that the smoothness of the functionals is larger than the ill–posedness of the problem, which is given by the polynomial decay rate of the characteristic function of the error. The limit distribution is a generalized Brownian bridge with a covariance structure that depends on the characteristic function of the error and on the functionals. The proposed estimators are optimal in the sense of semiparametric efficiency. The class of linear functionals is wide enough to incorporate the estimation of distribution functions. The proofs are based on smoothed empirical processes and mapping properties of the deconvolution operator.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 2486-2518.

Dates
First available in Project Euclid: 4 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1357307947

Digital Object Identifier
doi:10.1214/12-EJS757

Mathematical Reviews number (MathSciNet)
MR3020273

Zentralblatt MATH identifier
1295.62034

Subjects
Primary: 62G05: Estimation 60F05: Central limit and other weak theorems

Keywords
Deconvolution Donsker theorem efficiency distribution function smoothed empirical processes Fourier multipliers

Citation

Söhl, Jakob; Trabs, Mathias. A uniform central limit theorem and efficiency for deconvolution estimators. Electron. J. Statist. 6 (2012), 2486--2518. doi:10.1214/12-EJS757. https://projecteuclid.org/euclid.ejs/1357307947


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References

  • Bickel, P. J. and Ritov, Y. (2003). Nonparametric estimators which can be “plugged-in”., Ann. Statist. 31 1033–1053.
  • Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1998)., Efficient and Adaptive Estimation for Semiparametric Models. Springer, New York.
  • 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.
  • Bourdaud, G., Lanza de Cristoforis, M. and Sickel, W. (2006). Superposition operators and functions of bounded $p$-variation., Rev. Mat. Iberoam. 22 455–487.
  • Butucea, C. and Comte, F. (2009). Adaptive estimation of linear functionals in the convolution model and applications., Bernoulli 15 69-98.
  • Dattner, I., Goldenshluger, A. and Juditsky, A. (2011). On deconvolution of distribution functions., The Annals of Statistics 39 2477-2501.
  • de la Peña, V. H. and Giné, E. (1999)., Decoupling: From Dependence to Independence. Springer, New York.
  • Dudley, R. M. (1992). Fréchet differentiability, $p$-variation and uniform Donsker classes., Ann. Probab. 20 1968–1982.
  • Dudley, R. M. (1999)., Uniform Central Limit Theorems. Cambridge University Press, Cambridge.
  • Edmunds, D. E. and Triebel, H. (1996)., Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge.
  • Fan, J. (1991a). Asymptotic Normality for Deconvolution Kernel Density Estimators., Sankhyā: The Indian Journal of Statistics 53 97-110.
  • Fan, J. (1991b). On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems., The Annals of Statistics 19 1257-1272.
  • Fan, Y. and Liu, Y. (1997). A note on asymptotic normality for deconvolution kernel density estimators., Sankhyā Ser. A 59 138–141.
  • Fan, J. and Truong, Y. K. (1993). Nonparametric regression with errors in variables., Ann. Statist. 21 1900–1925.
  • Giné, E. and Nickl, R. (2008). Uniform central limit theorems for kernel density estimators., Probab. Theory Related Fields 141 333–387.
  • Giné, E. and Nickl, R. (2009). Uniform limit theorems for wavelet density estimators., Ann. Probab. 37 1605–1646.
  • Girardi, M. and Weis, L. (2003). Operator-valued Fourier multiplier theorems on Besov spaces., Mathematische Nachrichten 251 34–51.
  • Goldenshluger, A. and Pereverzev, S. V. (2003). On adaptive inverse estimation of linear functionals in Hilbert scales., Bernoulli 9 783-807.
  • Hall, P. and Lahiri, S. N. (2008). Estimation of distributions, moments and quantiles in deconvolution problems., Ann. Statist. 36 2110-2134.
  • Klenke, A. (2007)., Probability Theory: A Comprehensive Course (Universitext), 1st ed. Springer, London.
  • Kosorok, M. R. (2008)., Introduction to Empirical Processes and Semiparametric Inference, 1st ed. Springer Series in Statistics. Springer, New York.
  • Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators., Ann. Statist. 39 201–231.
  • Neumann, M. H. (1997). On the effect of estimating the error density in nonparametric deconvolution., J. Nonparametr. Statist. 7 307–330.
  • Nickl, R. (2006). Empirical and Gaussian processes on Besov classes. In, High dimensional probability. IMS Lecture Notes Monogr. Ser. 51 185–195.
  • 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.
  • Nickl, R. and Reiß, M. (2012). A Donsker theorem for Lévy measures., J. Funct. Anal. 263 3306–3332.
  • Qui, B. H. (1981). Bernstein’s theorem and translation invariant operators., Hiroshima Math. J. 11 81–96.
  • Radulović, D. and Wegkamp, M. (2000). Weak convergence of smoothed empirical processes: beyond Donsker classes. In, High dimensional probability, II. Progr. Probab. 47 89–105. Birkhäuser, Boston.
  • Schmidt–Hieber, J., Munk, A. and Dümbgen, L. (2012). Multiscale Methods for Shape Constraints in Deconvolution: Confidence Statements for Qualitative Features. arXiv, 1107.1404.
  • Triebel, H. (2010)., Theory of Function Spaces. Birkhäuser Verlag, Basel Reprint of the 1983 Edition.
  • van der Vaart, A. W. (1998)., Asymptotic statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.
  • van der Vaart, A. W. and Wellner, J. A. (1996)., Weak convergence and empirical processes. Springer Series in Statistics. Springer, New York.
  • van Es, B. and Uh, H.-W. (2005). Asymptotic normality of kernel-type deconvolution estimators., Scand. J. Statist. 32 467–483.
  • van Rooij, A. C. M., Ruymgaart, F. H. and van Zwet, W. R. (1999). Asymptotic efficiency of inverse estimators., Teor. Veroyatnost. i Primenen. 44 826–844. Translation in Theory Probab. Appl. 44(4), 722–738.