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Asymptotic normality of kernel-type deconvolution estimators. (English) Zbl 1089.62039

The PDF \(f\) of a random variable \(Y\) is estimated by i.i.d. observations \(X_i=Y_i+Z_i\), where \(Z_i\) is independent of \(Y_i\) and the PDF of \(Z\) is known. The asymptotic normality of the deconvolution-kernel-type estimate \(f_n(x)\) is demonstrated in the case where the characteristic function of \(Z_i\) behaves like \( C| t| ^{\lambda_0}\exp(-| t| ^\lambda/\mu) \) as \(| t| \to\infty\) (the Gaussian distribution corresponds to \(\lambda=\mu=2\), \(\lambda_0=0\)). The asymptotic variance of the normalized \(f_n\) can be independent of \(f\) and \(x\). The asymptotic normality of \(\int_a^b f_n(x)dx\) as an estimator of \(P\{a<Y_i\leq b\}\) is also derived. Results of simulations are presented.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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