Deconvolution of Cumulative Distribution Function with Unknown Noise Distribution

Abstract

Let \(Y\), \(X\) and \(\varepsilon \) be continuous univariate random variables satisfying the model \(Y = X + \varepsilon \). Herein \(X\) is of interest, \(Y\) is a noisy version of \(X\), and \(\varepsilon \) is a random noise independent of \(X\). This paper is devoted to a nonparametric estimation of cumulative distribution function \(F_{X}\) of \(X\) on the basis of independent random samples \((Y_{1}, \ldots , Y_{n})\) and \((\varepsilon '_{1}, \ldots , \varepsilon '_{m})\) drawn from the distributions of \(Y\) and \(\varepsilon \), respectively. We provide an estimator for \(F_{X}\) based on a direct inversion formula and the ridge-parameter regularization. Our estimator is shown to be mean consistency with respect to the mean squared error whenever the set of all zeros of the characteristic function of \(\varepsilon \) has Lebesgue measure zero. We then derive some convergence rates of the mean squared error uniformly on a nonparametric class for \(F_{X}\) and on some different regular classes for the density of \(\varepsilon \). A numerical example is performed to illustrate the efficiency of our method.

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Acknowledgements

We would like to thank the reviewers for fruitful comments and suggestions which help to significantly improve the paper.

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.321.

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Correspondence to Cao Xuan Phuong.

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Phuong, C.X. Deconvolution of Cumulative Distribution Function with Unknown Noise Distribution. Acta Appl Math (2020). https://doi.org/10.1007/s10440-020-00343-9

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Keywords

  • Cumulative distribution function
  • Non-standard noise
  • Mean consistency
  • Convergence rate

Mathematics Subject Classification (2010)

  • 62G07
  • 62G20