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Johannes, Jan

Deconvolution with unknown error distribution. (English) Zbl 1173.62018

Ann. Stat. 37, No. 5A, 2301-2323 (2009).
Summary: We consider the problem of estimating a density \(f_X\) using a sample \(Y_1,\dots, Y_n\) from \(f_Y=f_X* f_\varepsilon\), where \(f_\varepsilon\) is an unknown density. We assume that an additional sample \(\varepsilon_1,\dots, \varepsilon_m\) from \(f_\varepsilon\) is observed. Estimators of \(f_X\) and its derivatives are constructed by using nonparametric estimators of \(f_Y\) and \(f_\varepsilon\) and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknown error density \(f_\varepsilon \), where it is assumed that \(f_X\) satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density, if the density \(f_X\) belongs to a Sobolev space \(H_\wp\) and \(f_\varepsilon\) is ordinary smooth or supersmooth.

Cited in 46 Documents

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords:

deconvolution; Fourier transform; kernel estimation; spectral cut off; Sobolev space; source condition; optimal rate of convergence; Rosenblatt-Parzen kernel estimator
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\textit{J. Johannes}, Ann. Stat. 37, No. 5A, 2301--2323 (2009; Zbl 1173.62018)
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