umu.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Estimation of quadratic density functionals under m-dependence
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.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, we study estimation of certain integral functionals of one or two densities with samples from stationary m-dependent sequences. We consider two types of U-statistic estimators for these functionals that are functions of the number of epsilon-close vector observations in the samples. We show that the estimators are consistent and obtain their rates of convergence under weak distributional assumptions. In particular, we propose estimators based on incomplete U-statistics which have favorable consistency properties even when m-dependence is the only dependence condition that can be imposed on the stationary sequences. The results can be used for divergence and entropy estimation, and thus find many applications in statistics and applied sciences.

Keyword [en]
quadratic density functional, entropy estimation, divergence estimation, stationary m-dependent sequences, Renyi entropy, incomplete U-statistics
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:umu:diva-79956OAI: oai:DiVA.org:umu-79956DiVA: diva2:645442
Available from: 2013-09-04 Created: 2013-09-04 Last updated: 2013-10-14Bibliographically 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

Open Access in DiVA

Estimation of quadratic density functionals under m-dependence(177 kB)64 downloads
File information
File name FULLTEXT01.pdfFile size 177 kBChecksum SHA-512
ccb92c9c2a55d525b39c3d9e86b5dacb0ce5cea88e43a0d51e68b25da3260fe376861b537fe19f885fb004a0d16e44b0f679ddf5c47e267be698f3fe8f2e0cdd
Type fulltextMimetype application/pdf

Other links

Extern länk

Authority records BETA

Källberg, DavidSeleznjev, Oleg

Search in DiVA

By author/editor
Källberg, DavidSeleznjev, Oleg
By organisation
Department of Mathematics and Mathematical Statistics
Probability Theory and Statistics

Search outside of DiVA

GoogleGoogle Scholar
Total: 64 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

urn-nbn

Altmetric score

urn-nbn
Total: 58 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf