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On the asymptotic normality of the L 1 ‐ and L 2 ‐errors in histogram density estimation
Author(s) -
Beirlant Jan,
Györfi László,
Lugosi Gábor
Publication year - 1994
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315594
Subject(s) - mathematics , histogram , asymptotic distribution , density estimation , statistics , partition (number theory) , combinatorics , mathematical analysis , estimator , artificial intelligence , computer science , image (mathematics)
The L 1 and L 2 ‐errors of the histogram estimate of a density f from a sample X 1 ,X 2 ,…,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 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).