A central limit theorem for measurements on the logarithmic scale and its application to dimension estimates

Manfred Denker; Aleksey Min

doi:10.1016/j.jmva.2007.03.005

A central limit theorem for measurements on the logarithmic scale and its application to dimension estimates

Manfred Denker, Aleksey Min

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Georg August Universität Göttingen

Research output: Contribution to journal › Article › peer-review

Abstract

We show consistency and asymptotic normality of certain estimators for expected exponential growth rates under i.i.d. observations. These statistical functionals are of the formT (F) = ∫ log ∫ h (x, y) F (d x) F (d y)and are applicable to dimension estimates (information dimension), entropy estimates and estimations of the growth rate of "generating" functions. We also give an affirmative answer to a question posed by Keller in 1997 [A new estimator for information dimension with standard errors and confidence intervals, Stochastic Process. Appl. 71(2):187-206] whether this estimator, specialized for dimension, is an alternative to standard procedures.

Original language	English
Pages (from-to)	665-683
Number of pages	19
Journal	Journal of Multivariate Analysis
Volume	99
Issue number	4
DOIs	https://doi.org/10.1016/j.jmva.2007.03.005
State	Published - Apr 2008

Keywords

Central limit theorem
Information dimension
Local dimension
U-statistics

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10.1016/j.jmva.2007.03.005

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abstract = "We show consistency and asymptotic normality of certain estimators for expected exponential growth rates under i.i.d. observations. These statistical functionals are of the formT (F) = ∫ log ∫ h (x, y) F (d x) F (d y)and are applicable to dimension estimates (information dimension), entropy estimates and estimations of the growth rate of {"}generating{"} functions. We also give an affirmative answer to a question posed by Keller in 1997 [A new estimator for information dimension with standard errors and confidence intervals, Stochastic Process. Appl. 71(2):187-206] whether this estimator, specialized for dimension, is an alternative to standard procedures.",

keywords = "Central limit theorem, Information dimension, Local dimension, U-statistics",

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note = "Funding Information: The research of Aleksey Min was supported by the Graduiertenkolleg “Str{\"o}mungsinstabilit{\"a}ten und Turbulenz” at the University of G{\"o}ttingen. The authors would like to thank two referees for their valuable comments.",

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AB - We show consistency and asymptotic normality of certain estimators for expected exponential growth rates under i.i.d. observations. These statistical functionals are of the formT (F) = ∫ log ∫ h (x, y) F (d x) F (d y)and are applicable to dimension estimates (information dimension), entropy estimates and estimations of the growth rate of "generating" functions. We also give an affirmative answer to a question posed by Keller in 1997 [A new estimator for information dimension with standard errors and confidence intervals, Stochastic Process. Appl. 71(2):187-206] whether this estimator, specialized for dimension, is an alternative to standard procedures.

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KW - Local dimension

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JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

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A central limit theorem for measurements on the logarithmic scale and its application to dimension estimates

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