Beirlant, Jan; Györfi, László; Lugosi, Gábor On the asymptotic normality of the \(L_ 1\)- and \(L_ 2\)-errors in histogram density estimation. (English) Zbl 0816.62037 Can. J. Stat. 22, No. 3, 309-318 (1994). 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)]. Cited in 1 ReviewCited in 16 Documents MSC: 62G20 Asymptotic properties of nonparametric inference 62G07 Density estimation 62H12 Estimation in multivariate analysis Keywords:density estimation; central limit theorem; asymptotic normality; histogram estimate; cubic partition; asymptotic variances; asymptotic null distribution; goodness-of-fit test; total variation distance Citations:Zbl 0727.62053; Zbl 0018.22503 PDFBibTeX XMLCite \textit{J. Beirlant} et al., Can. J. Stat. 22, No. 3, 309--318 (1994; Zbl 0816.62037) Full Text: DOI References: [1] Bartlett, The characteristic function of a conditional statistic, J. London Math. Soc 13 pp 62– (1938) · Zbl 0018.22503 [2] Csörgö, Central limit theorems for Lp-norms of density estimators, Probab. Theory Related Fields 80 pp 269– (1988) [3] Csörgö, Strong Approximations in Probability and Statistics (1981) [4] Devroye, Nonparametric Density Estimation: The L1-View (1985) [5] Devroye, The kernel estimate is relatively stable, Probab. Theory Related Fields 77 pp 521– (1988) · Zbl 0627.62037 [6] Devroye, Nonparametric Functional Estimation and Related Topics pp 31– (1991) · doi:10.1007/978-94-011-3222-0_3 [7] Feller, An Introduction to Probability Theory and Its Applications II (1971) · Zbl 0219.60003 [8] Györfi, Nonparametric Functional Estimation and Related Topics pp 631– (1991) · doi:10.1007/978-94-011-3222-0_47 [9] Hall, Central limit theorem for integrated square error of multivariate nonparametric density estimators, J. Multivariate Anal. 14 pp 1– (1984) · Zbl 0528.62028 [10] Hoist, Asymptotic normality of sum-functions of spacings, Ann. Probab. 7 pp 1066– (1979) [11] Horváth, On Lp-norms of multivariates density estimators, Ann. Statist. 19 pp 1933– (1991) [12] LeCam, Un théorème sur la division d’un intervalle par des points pris au hasard, Publ. Inst. Statist. Univ. Paris 7 pp 7– (1958) [13] Rao, Statistical Inference and its Applications (1973) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.