• Bernoulli
  • Volume 5, Number 5 (1999), 907-925.

On pointwise adaptive nonparametric deconvolution

Alexander Goldenshluger

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We consider estimating an unknown function f from indirect white noise observations with particular emphasis on the problem of nonparametric deconvolution. Nonparametric estimators that can adapt to unknown smoothness of f are developed. The adaptive estimators are specified under two sets of assumptions on the kernel of the convolution transform. In particular, kernels having the Fourier transform with polynomially and exponentially decaying tails are considered. It is shown that the proposed estimates possess, in a sense, the best possible abilities for pointwise adaptation.

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Bernoulli, Volume 5, Number 5 (1999), 907-925.

First available in Project Euclid: 12 February 2007

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adaptive estimation deconvolution rates of convergence


Goldenshluger, Alexander. On pointwise adaptive nonparametric deconvolution. Bernoulli 5 (1999), no. 5, 907--925.

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  • [1] Abramovich, F. and Silverman, B. (1998) Wavelet decomposition approaches to statistical inverse problems. Biometrika, 85, 115-129.
  • [2] Anderssen, R.S. (1980) On the use of linear functionals for Abel-type integral equations. In R. Anderssen, F. De Hoog and M. Lucas (eds), The Application and Numerical Solution of Integral Equations. Amsterdam: Sijthoff and Noordhof International Publishers.
  • [3] Carrol, R.J. and Hall, P. (1988) Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc., 83, 1184-1186.
  • [4] Donoho, D.L. (1995) Non-linear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal., 2, 101-126.
  • [5] Efromovich, S. (1997) Robust and efficient recovery of a signal passed through a filter and then contaminated by non-Gaussian noise. IEEE Trans. Inform. Theory, 43, 1184-1191.
  • [6] Fan, J. (1991a) Global behavior of deconvolution kernel estimates. Statist. Sinica, 19, 541-551.
  • [7] Fan, J. (1991b) On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist., 19, 1257-1272.
  • [8] Goldberg, M. (1979) A method of adjoints for solving some ill-posed equations of the first kind. Appl. Math. Comput., 5, 123-130.
  • [9] Goldenshluger, A. and Nemirovski, A. (1997) On spatially adaptive estimation of nonparametric regression. Math. Methods Statist., 6(2), 135-170.
  • [10] Hirschmann, I. and Widder, D. (1955) The Convolution Transform. Princeton, NJ: Princeton University Press.
  • [11] Korostelev, A. and Tsybakov, A. (1993) Minimax Theory of Image Reconstruction, Lecture Notes in Statist. 82. New York: Springer-Verlag.
  • [12] Lepskii, O. (1991) Asymptotically minimax adaptive estimation I: Upper bounds. Optimally adaptive estimates. Theory Probab. Appl., 36, 682-697.
  • [13] Lepskii, O. (1992) Asymptotically minimax adaptive estimation II: Schemes without optimal adaptation. Adaptive estimators. Theory Probab. Appl., 37, 433-448.
  • [14] Lepskii, O., Mammen, E. and Spokoiny, V. (1997) Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. Ann. Statist., 25(3), 929-947.
  • [15] Liptser, R. and Shiryayev, A. (1977) Statistics of Random Processes I. New York: Springer-Verlag.
  • [16] Mair, B. and Ruymgaart, F.H. (1996) Statistical inverse estimation in Hilbert scale. SIAM J. Appl. Math., 56, 1424-1444.
  • [17] Masry, E. (1991) Multivariate probability density deconvolution for stationary random processes. IEEE Trans. Inform. Theory, 37, 1105-1115.
  • [18] Nychka, D. and Cox, D.D. (1989) Convergence rates for regularized solutions of integral equations from discrete noisy data. Ann. Statist., 17, 556-572.
  • [19] O'Sullivan, F. (1986) A statistical perspective on ill-posed inverse problems. Statist. Sci., 1, 502-527.
  • [20] Stefanski, L. and Carrol, R.J. (1990) Deconvoluting kernel density estimators. Statistics, 21, 169-184.
  • [21] Wahba, G. (1990) Spline Models for Observational Data. Philadelphia: Society for Industrial and Applied Mathematics.