Holzmann, Hajo; Boysen, Leif Integrated square error asymptotics for supersmooth deconvolution. (English) Zbl 1164.62335 Scand. J. Stat. 33, No. 4, 849-860 (2006). 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. Reviewer: R. E. Maiboroda (Kyïv) Cited in 6 Documents MSC: 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62E20 Asymptotic distribution theory in statistics 62G30 Order statistics; empirical distribution functions Keywords:degenerate U statistics; inverse problem; density estimation PDFBibTeX XMLCite \textit{H. Holzmann} and \textit{L. Boysen}, Scand. J. Stat. 33, No. 4, 849--860 (2006; Zbl 1164.62335) Full Text: DOI References: [1] Bickel P., Ann. Statist. 15 pp 513– (1987) [2] Bickel P., Ann. Statist. 1 pp 1071– (1973) [3] Billingsley P., Probability and measure (1995) · Zbl 0822.60002 [4] Butucea C., SORT 28 pp 9– (2004) [5] DOI: 10.2307/2290153 · Zbl 0673.62033 · doi:10.2307/2290153 [6] Denker M., Asymptotic distribution theory in nonparametric statistics (1985) · Zbl 0619.62019 · doi:10.1007/978-3-663-14229-4 [7] Fan J., Ann. Statist. 19 pp 1257– (1991) [8] Fan J., Sankhya. Ser, A 53 pp 97– (1991) [9] Fan J., Can. J. Statist. 20 pp 155– (1992) [10] DOI: 10.1016/0047-259X(84)90044-7 · Zbl 0528.62028 · doi:10.1016/0047-259X(84)90044-7 [11] DOI: 10.1007/BF00535267 · Zbl 0556.62020 · doi:10.1007/BF00535267 [12] Holzmann H., Integrated square error asymptotics for supersmooth deconvolution: technical details (2006) · Zbl 1164.62335 [13] Holzmann H., J. Multi variate Anal (2006) [14] Kotz S., Multivariate t distributions and their applications (2004) · Zbl 1100.62059 · doi:10.1017/CBO9780511550683 [15] DOI: 10.1137/S0036139994264476 · Zbl 0864.62020 · doi:10.1137/S0036139994264476 [16] Parzen E., Ann. Math. Statist. 33 pp 1065– (1962) [17] Piterbarg V. I., Math. Methods Statist. 2 pp 30– (1993) [18] Rosenblatt M., Ann. Math. Statist. 27 pp 832– (1956) [19] Stefanski L., Statistics 21 pp 169– (1990) [20] DOI: 10.1080/10485250310001644574 · Zbl 1216.62074 · doi:10.1080/10485250310001644574 [21] DOI: 10.1111/j.1467-9469.2005.00443.x · Zbl 1089.62039 · doi:10.1111/j.1467-9469.2005.00443.x [22] Zhang C.-H., Ann. Statist. 18 pp 806– (1990) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.