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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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##### References:
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