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Density Deconvolution with Small Berkson Errors

Abstract

The present paper studies density deconvolution in the presence of small Berkson errors, in particular, when the variances of the errors tend to zero as the sample size grows. It is known that when the Berkson errors are present, in some cases, the unknown density estimator can be obtained by simple averaging without using kernels. However, this may not be the case when Berkson errors are asymptotically small. By treating the former case as a kernel estimator with the zero bandwidth, we obtain the optimal expressions for the bandwidth. We show that the density of Berkson errors acts as a regularizer, so that the kernel estimator is unnecessary when the variance of Berkson errors lies above some threshold that depends on the shapes of the densities in the model and the number of observations.

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Acknowledgments

Marianna Pensky and Ramchandra Rimal were partially supported by National Science Foundation (NSF), grants DMS-1407475 and DMS-1712977. The authors also thank Alexander Tsybakov for the help with the proof of Lemma 3.

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Correspondence to R. Rimal or M. Pensky.

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Rimal, R., Pensky, M. Density Deconvolution with Small Berkson Errors. Math. Meth. Stat. 28, 208–227 (2019). https://doi.org/10.3103/S1066530719030025

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  • DOI: https://doi.org/10.3103/S1066530719030025

Keywords

  • density deconvolution
  • Berkson errors
  • bandwidth

AMS 2010 Subject Classification

  • 62G07
  • 62G20