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Deconvolution of a Cumulative Distribution Function with Some Non-standard Noise Densities

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Abstract

Let X be a continuous random variable having an unknown cumulative distribution function F. We study the problem of estimating F based on i.i.d. observations of a continuous random variable Y from the model Y = X + Z. Here, Z is a random noise distributed with known density g and is independent of X. We focus on some cases of g in which its Fourier transform can vanish on a countable subset of ℝ. We propose an estimator \(\hat F\) for F and then investigate upper bounds on convergence rate of \(\hat F\) under the root mean squared error. Some numerical experiments are also provided.

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Acknowledgements

We would like to thank the reviewers for their kind and careful reading of the paper and for helpful comments and suggestions which led to this improved version.

Funding

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.26.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City, No. 227 Nguyen Van Cu Street, Ward 4, District 5, Ho Chi Minh City, Vietnam

    Dang Duc Trong

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, No. 19 Nguyen Huu Tho Street, Tan Phong Ward, District 7, Ho Chi Minh City, Vietnam

    Cao Xuan Phuong

Authors
  1. Dang Duc Trong
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  2. Cao Xuan Phuong
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Correspondence to Cao Xuan Phuong.

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Trong, D.D., Phuong, C.X. Deconvolution of a Cumulative Distribution Function with Some Non-standard Noise Densities. Vietnam J. Math. 47, 327–353 (2019). https://doi.org/10.1007/s10013-018-0308-9

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  • DOI: https://doi.org/10.1007/s10013-018-0308-9

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