Almost sure \(L_ 1\)-norm convergence for data-based histogram density estimates
type: Item
instance of: scholarly article
title: Almost sure \(L_ 1\)-norm convergence for data-based histogram density estimates
zbMATH Open document ID: 0614.62044
DOI: 10.1016/0047-259X(87)90106-0
published in: Journal of Multivariate Analysis
publication date: 1987-01-01T00:00:00Z
review text: Let F be a one-dimensional distribution function with density f, and let \(X_ 1,...,X_ n\) denote an i.i.d. sample drawn from F. It is shown that two conditions of a general nature on the random grids imply strong \(L_ 1\)-consistency of the data-based histogram density estimator \(f_ n(x)=f_ n(x;X_ 1,...,X_ n)\), i.e. \[ \int^{\infty}_{- \infty}| f_ n(x)-f(x)| dx\to 0\quad a.s.\quad as\quad n\to \infty. \] The generalization of the result to the multidimensional case is also indicated.
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F25
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 3994762
zbMATH Keywords: mean absolute deviation
zbMATH Keywords: L sup 1 consistency
zbMATH Keywords: data-based histogram density estimator
author: Xi-Ru Chen
author: Lincheng Zhao
MaRDI profile type: Publication
cites work: Probability inequalities for empirical processes and a law of the iterated logarithm
cites work: Consistent cross-validated density estimation
cites work: Q3326516
cites work: Q3868610
cites work: An approximation of partial sums of independent RV'-s, and the sample DF. I
cites work: On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
cites work: An inequality concerning the deviation between theoretical and empirical distributions
description: scientific article; zbMATH DE number 3994762