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Statistical inference for Rényi entropy functionals
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
School of Mathematics, Cardiff University, Cardiff, UK.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0003-4116-3888
2012 (English)In: Conceptual modelling and its theoretical foundations / [ed] Antje Düsterhöft, Meike Klettke, Klaus-Dieter Schewe, Springer Berlin/Heidelberg, 2012, 36-51 p.Chapter in book (Refereed)
Abstract [en]

Numerous entropy-type characteristics (functionals) generalizing Rényi entropy are widely used in mathematical statistics, physics, information theory, and signal processing for characterizing uncertainty in probability distributions and distribution identification problems. We consider estimators of some entropy (integral) functionals for discrete and continuous distributions based on the number of epsilon-close vector records in the corresponding independent and identically distributed samples from two distributions. The proposed estimators are generalized U-statistics. We show the asymptotic properties of these estimators (e.g., consistency and asymptotic normality). The results can be applied in various problems in computer science and mathematical statistics (e.g., approximate matching for random databases, record linkage, image matching).

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2012. 36-51 p.
, Lecture Notes in Computer Science, ISSN 0302-9743 ; 7260/2012
Keyword [en]
entropy estimation, Rényi entropy, U-statistics, approximate matching, asymptotic normality
National Category
Probability Theory and Statistics
URN: urn:nbn:se:umu:diva-60788DOI: 10.1007/978-3-642-28279-9_5ISBN: 978-3-642-28278-2ISBN: 978-3-642-28279-9OAI: diva2:563154
Available from: 2012-11-14 Created: 2012-10-29 Last updated: 2016-05-27Bibliographically approved
In thesis
1. Nonparametric Statistical Inference for Entropy-type Functionals
Open this publication in new window or tab >>Nonparametric Statistical Inference for Entropy-type Functionals
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Icke-parametrisk statistisk inferens för entropirelaterade funktionaler
Abstract [en]

In this thesis, we study statistical inference for entropy, divergence, and related functionals of one or two probability distributions. Asymptotic properties of particular nonparametric estimators of such functionals are investigated. We consider estimation from both independent and dependent observations. The thesis consists of an introductory survey of the subject and some related theory and four papers (A-D).

In Paper A, we consider a general class of entropy-type functionals which includes, for example, integer order Rényi entropy and certain Bregman divergences. We propose U-statistic estimators of these functionals based on the coincident or epsilon-close vector observations in the corresponding independent and identically distributed samples. We prove some asymptotic properties of the estimators such as consistency and asymptotic normality. Applications of the obtained results related to entropy maximizing distributions, stochastic databases, and image matching are discussed.

In Paper B, we provide some important generalizations of the results for continuous distributions in Paper A. The consistency of the estimators is obtained under weaker density assumptions. Moreover, we introduce a class of functionals of quadratic order, including both entropy and divergence, and prove normal limit results for the corresponding estimators which are valid even for densities of low smoothness. The asymptotic properties of a divergence-based two-sample test are also derived.

In Paper C, we consider estimation of the quadratic Rényi entropy and some related functionals for the marginal distribution of a stationary m-dependent sequence. We investigate asymptotic properties of the U-statistic estimators for these functionals introduced in Papers A and B when they are based on a sample from such a sequence. We prove consistency, asymptotic normality, and Poisson convergence under mild assumptions for the stationary m-dependent sequence. Applications of the results to time-series databases and entropy-based testing for dependent samples are discussed.

In Paper D, we further develop the approach for estimation of quadratic functionals with m-dependent observations introduced in Paper C. We consider quadratic functionals for one or two distributions. The consistency and rate of convergence of the corresponding U-statistic estimators are obtained under weak conditions on the stationary m-dependent sequences. Additionally, we propose estimators based on incomplete U-statistics and show their consistency properties under more general assumptions.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2013. 21 p.
entropy estimation, Rényi entropy, divergence estimation, quadratic density functional, U-statistics, consistency, asymptotic normality, Poisson convergence, stationary m-dependent sequence, inter-point distances, entropy maximizing distribution, two-sample problem, approximate matching
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
urn:nbn:se:umu:diva-79976 (URN)978-91-7459-701-1 (ISBN)
Public defence
2013-09-27, MIT-huset, MA121, Umeå universitet, Umeå, 10:00 (English)
Available from: 2013-09-06 Created: 2013-09-04 Last updated: 2013-09-05Bibliographically approved

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Källberg, DavidOleg, Seleznjev
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