On the asymptotic normality of the L1‐ and L2‐errors in histogram density estimation

Jan Beirlant, L. Györfi, Gábor Lugosi

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The L1 and L2‐errors of the histogram estimate of a density f from a sample X1,X2,…,Xn 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 Györfi and van der Meulen (1991). This note uses the idea of partial inversion for obtaining characteristic functions of conditional distributions, which goes back at least to Bartlett (1938).

Original languageEnglish
Pages (from-to)309-318
Number of pages10
JournalCanadian Journal of Statistics
Issue number3
Publication statusPublished - 1994



  • central limit theorem
  • histogram estimate.
  • Nonparametric density estimation
  • Primary 62H12
  • secondary 62G05.

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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