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StrongL 1-norm consistency of data based histogram estimates of densities

StrongL 1-norm consistency of data based histogram estimates of densities In this paper we investigate the problem of estimating thed-dimensional probability densityf(x),x∈R d from a sample of sizen. The non-parametric estimator is the data based histogramf n (x) as defined in (1). Under suitable conditions, we have proved theL 1-norm consistance of this estimate, that is $$\mathop {\lim }\limits_{n \to \infty } \int_{R^a } {|f(x) - f_n (x)|dx = 0,{\text{ a}}{\text{.s}}{\text{.}}}$$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

StrongL 1-norm consistency of data based histogram estimates of densities

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Publisher
Springer Journals
Copyright
Copyright © 1985 by Science Press and D. Reidel Publishing Company
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF01665849
Publisher site
See Article on Publisher Site

Abstract

In this paper we investigate the problem of estimating thed-dimensional probability densityf(x),x∈R d from a sample of sizen. The non-parametric estimator is the data based histogramf n (x) as defined in (1). Under suitable conditions, we have proved theL 1-norm consistance of this estimate, that is $$\mathop {\lim }\limits_{n \to \infty } \int_{R^a } {|f(x) - f_n (x)|dx = 0,{\text{ a}}{\text{.s}}{\text{.}}}$$

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Apr 25, 2005

References