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

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).