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On the asymptotic normality of the \(L_ 1\)- and \(L_ 2\)-errors in histogram density estimation. (English) Zbl 0816.62037
Summary: The \(L_ 1\)- and \(L_ 2\)-errors of the histogram estimate of a density \(f\) from a sample \(X_ 1,X_ 2,\dots,X_ n\) using a cubic partition are shown to be asymptotically normal without any unnecessary conditions imposed on the density \(f\). The asymptotic variances are shown to depend on \(f\) only through the corresponding norm of \(f\). From this follows the asymptotic null distribution of a goodness-of-fit test based on the total variation distance, introduced by L. Györfi and E. C. van der Meulen [Nonparametric functional estimation and related topics, NATO ASI Ser., Ser. C 335, 631-645 (1991; Zbl 0727.62053)]. This note uses the idea of partial inversion for obtaining characteristic functions of conditional distributions, which goes back at least to M. S. Bartlett [J. Lond. Math. Soc. 13, 62-67 (1938; Zbl 0018.22503)].

MSC:
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
62H12 Estimation in multivariate analysis
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