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Integrated square error asymptotics for supersmooth deconvolution. (English) Zbl 1164.62335

The PDF of the r.v. \(Z\) is estimated by i.i.d. noisy observations \(X_j =Z_j+\varepsilon_j\), \(j=1,\dots,n\), where \(Z_j\) and \(\varepsilon_j\) are independent, and the distribution of \(\varepsilon_j\) is known. The authors consider the deconvolution kernel estimate \[ \hat f_n(x)=(2\pi)^{-1}\int_R e^{itx}\Phi_K(ht)(\hat\Phi_n(t)/\Phi_\varepsilon(t))dt, \] where \(\hat\Phi_n(t)\) is the empirical characteristic function of \(X_j\), \(\Phi_K\) is the kernel \(K\) Fourier transform, \(h\) is the bandwidth, \(\Phi_\varepsilon\) is the characteristic function of \(\varepsilon\). The asymptotic behaviour of the integrated square error \[ T_n=\int_R(\hat f_n(x)-E\hat f_N(x))^2dx \] is investigated in the supersmooth case, i.e., when \(\Psi_\varepsilon(t)\sim C| t| ^{\lambda_0}e^{-| t| ^\lambda\mu}\) as \(| t| \to\infty\). It is shown that the limit distribution of normalized \(T_n\) is \(\chi^2\) with 2 degrees of freedom and the normalizing sequence does not depend on \(f\). Results of simulations are presented.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
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