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Estimation of entropy-type integral functionals
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2016 (English)In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 45, no 4, 887-905 p.Article in journal (Other academic) Published
Abstract [en]

Entropy-type integral functionals of densities are widely used in mathematical statistics, information theory, and computer science. Examples include measures of closeness between distributions (e.g., density power divergence) and uncertainty characteristics for a random variable (e.g., Renyi entropy). In this paper, we study U-statistic estimators for a class of such functionals. The estimators are based on ε-close vector observations in the corresponding independent and identically distributed samples. We prove asymptotic properties of the estimators (consistency and asymptotic normality) under mild integrability and smoothness conditions for the densities. The results can be applied in diverse problems in mathematical statistics and computer science (e.g., distribution identication problems, approximate matching for random databases, two-sample problems).

Place, publisher, year, edition, pages
2016. Vol. 45, no 4, 887-905 p.
Keyword [en]
Divergence estimation, asymptotic normality, U-statistics, inter-point distances, quadratic functional, entropy estimation
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:umu:diva-60993DOI: 10.1080/03610926.2013.853789ISI: 000370612900005OAI: oai:DiVA.org:umu-60993DiVA: diva2:565242
Available from: 2012-11-06 Created: 2012-11-06 Last updated: 2017-12-07Bibliographically 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.
Keyword
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
Identifiers
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)
Opponent
Supervisors
Available from: 2013-09-06 Created: 2013-09-04 Last updated: 2013-09-05Bibliographically approved

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Källberg, DavidSeleznjev, Oleg

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