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Vietnam Journal of Mathematics

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

Ridge-Parameter Regularization to Deconvolution Problem with Unknown Error Distribution

  • Dang Duc Trong
  • Cao Xuan Phuong
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.

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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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