The Annals of Statistics

A ridge-parameter approach to deconvolution

Peter Hall and Alexander Meister

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Abstract

Kernel methods for deconvolution have attractive features, and prevail in the literature. However, they have disadvantages, which include the fact that they are usually suitable only for cases where the error distribution is infinitely supported and its characteristic function does not ever vanish. Even in these settings, optimal convergence rates are achieved by kernel estimators only when the kernel is chosen to adapt to the unknown smoothness of the target distribution. In this paper we suggest alternative ridge methods, not involving kernels in any way. We show that ridge methods (a) do not require the assumption that the error-distribution characteristic function is nonvanishing; (b) adapt themselves remarkably well to the smoothness of the target density, with the result that the degree of smoothness does not need to be directly estimated; and (c) give optimal convergence rates in a broad range of settings.

Article information

Source
Ann. Statist., Volume 35, Number 4 (2007), 1535-1558.

Dates
First available in Project Euclid: 29 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1188405621

Digital Object Identifier
doi:10.1214/009053607000000028

Mathematical Reviews number (MathSciNet)
MR2351096

Zentralblatt MATH identifier
1147.62031

Subjects
Primary: 62G07: Density estimation 62F05: Asymptotic properties of tests

Keywords
cross-validation density deconvolution errors in variables kernel methods minimax optimality optimal convergence rates smoothing-parameter choice

Citation

Hall, Peter; Meister, Alexander. A ridge-parameter approach to deconvolution. Ann. Statist. 35 (2007), no. 4, 1535--1558. doi:10.1214/009053607000000028. https://projecteuclid.org/euclid.aos/1188405621


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References

  • Berkson, J. (1950). Are there two regressions? J. Amer. Statist. Assoc. 45 164–180.
  • Buonaccorsi, J. P. and Lin, C.-D. (2002). Berkson measurement error in designed repeated measures studies with random coefficients. J. Statist. Plann. Inference 104 53–72.
  • Butucea, C. and Tsybakov, A. B. (2007). Sharp optimality for density deconvolution with dominating bias. Theory Probab. Appl. To appear.
  • Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.
  • Carroll, R. J., Ruppert, D. and Stefanski, L. A. (1995). Measurement Error in Nonlinear Models. Chapman and Hall, London.
  • Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006). Penalized contrast estimator for adaptive density deconvolution. Manuscript. Available at fr.arxiv.org/abs/math.st/0601091.
  • Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006). Finite sample penalization in adaptive density deconvolution. Manuscript. Available at fr.arxiv.org/abs/math.st/0601098.
  • Delaigle, A. and Gijbels, I. (2002). Estimation of integrated squared density derivatives from a contaminated sample. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 869–886.
  • Delaigle, A. and Gijbels, I. (2004). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249–267.
  • Delaigle, A. and Gijbels, I. (2004). Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Ann. Inst. Statist. Math. 56 19–47.
  • Devroye, L. (1989). Consistent deconvolution in density estimation. Canad. J. Statist. 17 235–239.
  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
  • Fan, J. (1991). Global behavior of deconvolution kernel estimates. Statist. Sinica 1 541–551.
  • Fan, J. (1992). Deconvolution with supersmooth distributions. Canad. J. Statist. 20 155–169.
  • Fan, J. (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 600–610.
  • Fan, J. and Koo, J.-Y. (2002). Wavelet deconvolution. IEEE Trans. Inform. Theory 48 734–747.
  • Fan, J. and Truong, Y. (1993). Nonparametric regression with errors in variables. Ann. Statist. 21 1900–1925.
  • Groeneboom, P. and Jongbloed, G. (2003). Density estimation in the uniform deconvolution model. Statist. Neerlandica 57 136–157.
  • Hall, P. and Meister, A. (2006). A ridge-parameter approach to deconvolution (long version). Available at www.ms.unimelb.edu.au/~halpstat/hmeirev-long-version.pdf.
  • Hesse, C. H. (1999). Data-driven deconvolution. J. Nonparametr. Statist. 10 343–373.
  • Hesse, C. H. and Meister, A. (2004). Optimal iterative density deconvolution. J. Nonparametr. Statist. 16 879–900.
  • Liu, M. C. and Taylor, R. L. (1989). A consistent nonparametric density estimator for the deconvolution problem. Canad. J. Statist. 17 427–438.
  • Meister, A. (2004). On the effect of misspecifying the error density in a deconvolution problem. Canad. J. Statist. 32 439–449.
  • Meister, A. (2006). Density estimation with normal measurement error with unknown variance. Statist. Sinica 16 195–211.
  • Neumann, M. H. (1997). On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Stat. 7 307–330.
  • Reeves, G. K., Cox, D. R., Darby, S. C. and Whitley, E. (1998). Some aspects of measurement error in explanatory variables for continuous and binary regression models. Stat. Med. 17 2157–2177.
  • Stefanski, L. A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229–235.
  • Stefanski, L. A. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169–184.
  • van Es, B., Spreij, P. and van Zanten, H. (2003). Nonparametric volatility density estimation. Bernoulli 9 451–465.
  • Zhang, C.-H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806–831.