Vietnam Journal of Mathematics

, Volume 43, Issue 2, pp 239–256 | Cite as

Ridge-Parameter Regularization to Deconvolution Problem with Unknown Error Distribution

Article

Abstract

Our aim in this article is to estimate a density function f of i.i.d. random variables X 1, … , X n from a noise model Y j = X j + Z j , j = 1, 2, … , n. Here, (Z j )1≤jn is independent of (X j )1≤jn and is a finite sequence of i.i.d. noise random variables distributed with an unknown density function g. This problem is known as the deconvolution problem in nonparametric statistics. The general case in which the error density function g is unknown and its Fourier transform g ft can vanish on a subset of ℝ has still not been considered much. In the present article, we consider this case. Using direct i.i.d. data \(Z^{\prime }_{1},\ldots , Z^{\prime }_{m}\) which are collected in separated independent experiments, we propose an estimator \(\hat g\) to the unknown density function g. After that, applying a ridge-parameter regularization method and an estimation of the Lebesgue measure of low level sets of g ft, we give an estimator \(\hat f\) to the target density function f and evaluate therateof convergence of the quantity \(\mathbb {E}\|\hat f - f\|_{2}^{2}\).

Keywords

Deconvolution Density function Fourier transform Estimator Mean integrated squared error 

Mathematics Subject Classification (2010)

62F12 62G07 

Notes

Acknowledgments

The authors would like to thank the referees for careful reading of the article and for helpful comments and suggestions leading to the improved version of our article.

References

  1. 1.
    Carroll, R.J., Hall, P.: Optimal rates of convergence for deconvolving a density. J. Am. Stat. Assoc. 83(404), 1184–1186 (1988)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Comte, F., Lacour, C.: Data-driven density estimation in the presence of additive noise with unknown distribution. J. R. Stat. Soc. Ser. B 73, 601–627 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Devroye, L.: Consistent deconvolution in density estimation. Can. J. Stat. 17, 235–239 (1989)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Diggle, P.J., Hall, P.: A Fourier approach to nonparametric deconvolution of a density estimate. J. R. Stat. Soc. Ser. B 55, 523–531 (1993)MATHMathSciNetGoogle Scholar
  5. 5.
    Fan, J.: On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Stat. 19, 1257–1272 (1991)CrossRefMATHGoogle Scholar
  6. 6.
    Fan, J.: Asymptotic normality for deconvolution kernel density estimators. Sankhya: Indian. J. Stat. Ser. A 53, 97–110 (1991)MATHGoogle Scholar
  7. 7.
    Fan, J.: Global behavior of deconvolution kernel estimates. Stat. Sin. 1, 541–551 (1991)MATHGoogle Scholar
  8. 8.
    Hall, P., Meister, A.: A ridge-parameter approach to deconvolution. Ann. Stat. 35, 1535–1558 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Johannes, J.: Deconvolution with unknown error distribution. Ann. Stat. 37, 2301–2323 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Levin, B.Y.: Lectures on Entire Functions. Trans. Math. Monographs, vol. 150. AMS, Providence, Rhole Island (1996)Google Scholar
  11. 11.
    Lounici, K., Nickl, R.: Global uniform risk bounds for wavelet deconvolution estimators. Ann. Stat. 39, 201–231 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Meister, A.: Non-estimability in spite of identifiability in density deconvolution. Math. Methods Stat. 14, 479–487 (2005)MathSciNetGoogle Scholar
  13. 13.
    Meister, A.: Deconvolution Problem in Nonparametric Statistics. Springer, Berlin (2009)CrossRefMATHGoogle Scholar
  14. 14.
    Meister, A., Neumann, M.H.: Deconvolution from non-standard error densities under replicated measurements. Stat. Sin. 20, 1609–1636 (2010)MATHMathSciNetGoogle Scholar
  15. 15.
    Neumann, M.H.: On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Stat. 7, 307–330 (1997)CrossRefMATHGoogle Scholar
  16. 16.
    Pensky, M., Vidakovic, B.: Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Stat. 27, 2033–2053 (1999)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Stefanski, L.A., Carroll, R.J.: Deconvoluting kernel density estimators. Stat. 21, 169–184 (1990)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Wang, X.F., Ye, D.: The effects of error magnitude and bandwidth selection for deconvolution with unknown error distribution. J. Nonparametr. Stat. 24, 153–167 (2012)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Zhang, C.H.: Fourier methods for estimating mixing densities and distributions. Ann. Stat. 18, 806–831 (1990)CrossRefMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceHo Chi Minh City National UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam

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