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Rank covariance matrix estimation of a partially known covariance matrix
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
2008 (English)In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 138, no 12, 3667-3673 p.Article in journal (Refereed) Published
Abstract [en]

Classical multivariate methods are often based on the sample covariance matrix, which is very sensitive to outlying observations. One alternative to the covariance matrix is the affine equivariant rank covariance matrix (RCM) that has been studied in Visuri et al. [2003. Affine equivariant multivariate rank methods.]. Statist. Plann. Inference 114, 161-185]. In this article we assume that the covariance matrix is partially known and study how to estimate the corresponding RCM. We use the properties that the RCM is affine equivariant and that the RCM is proportional to the inverse of the regular covariance matrix, and hence reduce the problem of estimating the original RCM to estimating marginal rank covariance matrices. This is a great computational advantage when the dimension of the original data vector is large.

Place, publisher, year, edition, pages
2008. Vol. 138, no 12, 3667-3673 p.
Keyword [en]
multivariate ranks, rank covariance matrix, marginal rank covariance matrix, elliptical distributions, affine equivariance
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-97603DOI: 10.1016/j.jspi.2007.11.015ISI: 000259755600004OAI: oai:DiVA.org:uu-97603DiVA: diva2:172612
Available from: 2008-10-10 Created: 2008-10-10 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Rank Estimation in Elliptical Models: Estimation of Structured Rank Covariance Matrices and Asymptotics for Heteroscedastic Linear Regression
Open this publication in new window or tab >>Rank Estimation in Elliptical Models: Estimation of Structured Rank Covariance Matrices and Asymptotics for Heteroscedastic Linear Regression
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis deals with univariate and multivariate rank methods in making statistical inference. It is assumed that the underlying distributions belong to the class of elliptical distributions. The class of elliptical distributions is an extension of the normal distribution and includes distributions with both lighter and heavier tails than the normal distribution.

In the first part of the thesis the rank covariance matrices defined via the Oja median are considered. The Oja rank covariance matrix has two important properties: it is affine equivariant and it is proportional to the inverse of the regular covariance matrix. We employ these two properties to study the problem of estimating the rank covariance matrices when they have a certain structure.

The second part, which is the main part of the thesis, is devoted to rank estimation in linear regression models with symmetric heteroscedastic errors. We are interested in asymptotic properties of rank estimates. Asymptotic uniform linearity of a linear rank statistic in the case of heteroscedastic variables is proved. The asymptotic uniform linearity property enables to study asymptotic behaviour of rank regression estimates and rank tests. Existing results are generalized and it is shown that the Jaeckel estimate is consistent and asymptotically normally distributed also for heteroscedastic symmetric errors.

Place, publisher, year, edition, pages
Uppsala: Universitetsbiblioteket, 2008. 42 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 58
Keyword
elliptical distributions, multivariate ranks, rank covariance matrix, linear rank regression, heteroscedastic errors, linear rank statistics
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-9305 (URN)978-91-506-2026-9 (ISBN)
Public defence
2008-11-03, Polhemsalen, Ångström Laboratory, Lägerhyddsvägen 1, Uppsala, 10:15
Opponent
Supervisors
Available from: 2008-10-10 Created: 2008-10-10 Last updated: 2012-07-26Bibliographically approved

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