Bernoulli

  • Bernoulli
  • Volume 11, Number 2 (2005), 309-340.

Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model

Cristina Butucea and Catherine Matias

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Abstract

We consider a semiparametric convolution model where the noise has known Fourier transform which decays asymptotically as an exponential with unknown scale parameter; the deconvolution density is less smooth than the noise in the sense that the tails of the Fourier transform decay more slowly, ensuring the identifiability of the model. We construct a consistent estimation procedure for the noise level and prove that its rate is optimal in the minimax sense. Two convergence rates are distinguished according to different smoothness properties for the unknown density. If the tail of its Fourier transform does not decay faster than exponentially, the asymptotic optimal rate and exact constant are evaluated, while if it does not decay faster than polynomially, this rate is evaluated up to a constant. Moreover, we construct a consistent estimator of the unknown density, by using a plug-in method in the classical kernel estimation procedure. We establish that the rates of estimation of the deconvolution density are slower than in the case of an entirely known noise distribution. In fact, nonparametric rates of convergence are equal to the rate of estimation of the noise level, and we prove that these rates are minimax. In a few specific cases the plug-in method converges at even slower rates.

Article information

Source
Bernoulli, Volume 11, Number 2 (2005), 309-340.

Dates
First available in Project Euclid: 17 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.bj/1116340297

Digital Object Identifier
doi:10.3150/bj/1116340297

Mathematical Reviews number (MathSciNet)
MR2132729

Zentralblatt MATH identifier
1063.62044

Keywords
analytic densities deconvolution L_2 risk minimax estimation noise level pointwise risk semiparametric model Sobolev classes supersmooth densities

Citation

Butucea, Cristina; Matias, Catherine. Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli 11 (2005), no. 2, 309--340. doi:10.3150/bj/1116340297. https://projecteuclid.org/euclid.bj/1116340297


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References

  • [1] Butucea, C. (2001) Exact adaptive pointwise estimation on Sobolev classes of densities, ESAIM Probab. Statist., 5, 1-31.
    Mathematical Reviews (MathSciNet): MR1845320
    Digital Object Identifier: doi:10.1051/ps:2001100
  • [2] Butucea, C. and Tsybakov, A. (2004) Sharp optimality for density deconvolution with dominating bias. arXiv: math.ST/0409471.
  • [3] Carroll, R.J. and Hall, P. (1988) Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc., 83, 1184-1186.
    Mathematical Reviews (MathSciNet): MR997599
    Zentralblatt MATH: 0673.62033
    Digital Object Identifier: doi:10.2307/2290153
    JSTOR: links.jstor.org
  • [4] Efromovich, S. (1997) Density estimation for the case of supersmooth measurement error. J. Amer. Statist. Assoc., 92, 526-535.
    Mathematical Reviews (MathSciNet): MR1467846
    Zentralblatt MATH: 0890.62027
    Digital Object Identifier: doi:10.2307/2965701
    JSTOR: links.jstor.org
  • [5] Goldenshluger, A. (1999) On pointwise adaptive nonparametric deconvolution, Bernoulli, 5, 907-925.
    Mathematical Reviews (MathSciNet): MR1715444
    Digital Object Identifier: doi:10.2307/3318449
    Project Euclid: euclid.bj/1171290404
  • [6] Goldenshluger, A. (2002) Density deconvolution in the circular structural model. J. Multivariate Anal., 81, 360-375.
    Mathematical Reviews (MathSciNet): MR1906385
    Zentralblatt MATH: 0998.62029
    Digital Object Identifier: doi:10.1006/jmva.2001.2015
  • [7] Gradshteyn, I. and Ryzhik, I. (2000) Table of Integrals, Series, and Products, 6th edition. San Diego, CA: Academic Press.
    Mathematical Reviews (MathSciNet): MR1773820
  • [8] Koltchinskii, V.I. (2000) Empirical geometry of multivariate data: a deconvolution approach, Ann. Statist., 28, 591-629.
    Mathematical Reviews (MathSciNet): MR1790011
    Zentralblatt MATH: 1105.62345
    Digital Object Identifier: doi:10.1214/aos/1016218232
    Project Euclid: euclid.aos/1016218232
  • [9] Lindsay, B.G. (1986) Exponential family mixture models (with least-squares estimators). Ann. Statist., 14, 124-137.
    Mathematical Reviews (MathSciNet): MR829558
    Zentralblatt MATH: 0587.62057
    Digital Object Identifier: doi:10.1214/aos/1176349845
    Project Euclid: euclid.aos/1176349845
  • [10] Matias, C. (2002) Semiparametric deconvolution with unknown noise variance, ESAIM Probab. Statist., 6, 271-292.
    Mathematical Reviews (MathSciNet): MR1943151
    Digital Object Identifier: doi:10.1051/ps:2002015
  • [11] van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes. New York: Springer-Verlag.
    Mathematical Reviews (MathSciNet): MR1385671
    Zentralblatt MATH: 0862.60002
  • [12] Zhang, C.-H. (1990) Fourier methods for estimating mixing densities and distributions, Ann. Statist., 18, 806-831.
    Mathematical Reviews (MathSciNet): MR1056338
    Zentralblatt MATH: 0778.62037
    Digital Object Identifier: doi:10.1214/aos/1176347627
    Project Euclid: euclid.aos/1176347627
  • [13] Zolotarev, V. (1986) One-Dimensional Stable Distributions, Transl. Math. Monogr. 65. Providence, RI: American Mathematical Society.
    Mathematical Reviews (MathSciNet): MR854867

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