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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Statistical estimation of measure invariants


Author: E. A. Timofeev
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 17 (2005), nomer 3.
Journal: St. Petersburg Math. J. 17 (2006), 527-551
MSC (2000): Primary 28A75, 62L20
Published electronically: March 21, 2006
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Abstract | References | Similar Articles | Additional Information

Abstract: New invariants of measures, called the $ \beta$-statentropy, are described. They are similar to the entropy and the $ HP$-spectrum for dimensions. The $ \beta$-statentropy admits construction of a statistical estimator calculated by $ n$ independent points distributed in accordance with a given measure. The accuracy of this estimator is $ \mathcal{O}(n^{-c})$, where $ c$ is some constant, and the complexity of calculation is $ \mathcal{O}(n^2)$.

It is shown that for an exact dimensional measure the 0-statentropy coincides with the Hausdorff dimension, and for a Markov measure the $ \beta$-statentropy coincides with the $ HP$-spectrum for dimensions.

An application of the $ \beta$-statentropy to finding the entropy and dimensional characteristics of dynamical systems is described.


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Additional Information

E. A. Timofeev
Affiliation: Yaroslavl′ State University, Sovetskaya Street 14, Yaroslavl′ 150000, Russia
Email: tim@uniyar.ac.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00919-8
PII: S 1061-0022(06)00919-8
Keywords: $\beta$-statentropy, statistical estimator, exact dimensional measure, Markov measure
Received by editor(s): February 20, 2004
Published electronically: March 21, 2006
Article copyright: © Copyright 2006 American Mathematical Society