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Dimension spectra for multifractal measures with connections to nonparametric density estimation

Frigyesi, Attila LU and Hössjer, Ola LU (2001) In Journal of Nonparametric Statistics 13(3). p.351-395
Abstract

We consider relations between Rényi's and Hentschel - Procaccia's definitions of generalized dimensions of a probability measure μ, and give conditions under which the two concepts are equivalent/different. Estimators of the dimension spectrum are developed, and strong consistency is established. Particular cases of our estimators are methods based on the sample correlation integral and box counting. Then we discuss the relation between generalized dimensions and kernel density estimators f̂. It was shown in Frigyesi and Hössjer (1998), that ∫ f̂1+q(x)dx diverges with increasing sample size and decreasing bandwidth if the marginal distribution μ has a singular part and q > 0. In this paper, we show that the rate of... (More)

We consider relations between Rényi's and Hentschel - Procaccia's definitions of generalized dimensions of a probability measure μ, and give conditions under which the two concepts are equivalent/different. Estimators of the dimension spectrum are developed, and strong consistency is established. Particular cases of our estimators are methods based on the sample correlation integral and box counting. Then we discuss the relation between generalized dimensions and kernel density estimators f̂. It was shown in Frigyesi and Hössjer (1998), that ∫ f̂1+q(x)dx diverges with increasing sample size and decreasing bandwidth if the marginal distribution μ has a singular part and q > 0. In this paper, we show that the rate of divergence depends on the qth generalized Rényi dimension of μ.

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Contribution to journal
publication status
published
subject
keywords
Box counting, Correlation integral, Fractal dimension estimation, Generalized dimensions, Hentschel-Proccacia dimension, Kernel density estimates, Rényi dimension
in
Journal of Nonparametric Statistics
volume
13
issue
3
pages
45 pages
publisher
Taylor & Francis
external identifiers
  • scopus:0347117696
ISSN
1048-5252
DOI
10.1080/10485250108832857
language
English
LU publication?
yes
id
f280e939-eb1c-4115-b203-95491eb862b9
date added to LUP
2019-06-14 22:37:46
date last changed
2020-01-13 02:01:34
@article{f280e939-eb1c-4115-b203-95491eb862b9,
  abstract     = {<p>We consider relations between Rényi's and Hentschel - Procaccia's definitions of generalized dimensions of a probability measure μ, and give conditions under which the two concepts are equivalent/different. Estimators of the dimension spectrum are developed, and strong consistency is established. Particular cases of our estimators are methods based on the sample correlation integral and box counting. Then we discuss the relation between generalized dimensions and kernel density estimators f̂. It was shown in Frigyesi and Hössjer (1998), that ∫ f̂<sup>1+q</sup>(x)dx diverges with increasing sample size and decreasing bandwidth if the marginal distribution μ has a singular part and q &gt; 0. In this paper, we show that the rate of divergence depends on the qth generalized Rényi dimension of μ.</p>},
  author       = {Frigyesi, Attila and Hössjer, Ola},
  issn         = {1048-5252},
  language     = {eng},
  month        = {01},
  number       = {3},
  pages        = {351--395},
  publisher    = {Taylor & Francis},
  series       = {Journal of Nonparametric Statistics},
  title        = {Dimension spectra for multifractal measures with connections to nonparametric density estimation},
  url          = {http://dx.doi.org/10.1080/10485250108832857},
  doi          = {10.1080/10485250108832857},
  volume       = {13},
  year         = {2001},
}