We consider the problem of nonparametric distribution function deconvolution with respect to the $1$-Wasserstein metric when the error distribution is unknown and ordinary smooth. We propose an estimator based on the integral of the classical deconvolution kernel density estimator, where the characteristic function of the error distribution is estimated by its empirical counterpart, using a sample of i.i.d. observations. Assuming H\"older or Sobolev regularity of the mixing density, the estimator achieves a rate that has the same structure as in the known error case, but is determined by the minimum of the error sample size and the recorded signal sample size.
Wasserstein deconvolution with unknown error distribution
C. Scricciolo
2025-01-01
Abstract
We consider the problem of nonparametric distribution function deconvolution with respect to the $1$-Wasserstein metric when the error distribution is unknown and ordinary smooth. We propose an estimator based on the integral of the classical deconvolution kernel density estimator, where the characteristic function of the error distribution is estimated by its empirical counterpart, using a sample of i.i.d. observations. Assuming H\"older or Sobolev regularity of the mixing density, the estimator achieves a rate that has the same structure as in the known error case, but is determined by the minimum of the error sample size and the recorded signal sample size.
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