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Location-invariant and non-invariant tests for large dimensional covariance matrices under normality and non-normality
Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
2014 (English)Report (Other academic)
Abstract [en]

Test statistics for homogeneity, sphericity and identity of high-dimensional covariance matrices are presented under a wide variety of very general conditions when the dimension of the vector, $p$, may exceed the sample size, $n_i$, $i = 1, \ldots, g$. First, location-invariant tests are presented under normality assumption, followed by their robustness to normality by replacing the normality assumption with a mild alternative multivariate model. The two types of tests are then presented in non-invariant form, again under normality and non-normality. Tests of homogeneity of covariance matrices in all cases are immediately supplemented by the tests for sphericity and identity of the common covariance matrix under the null hypothesis. Both location-invariant and non-invariant tests are composed of estimators that are defined as $U$-statistics with kernels of different degrees. Hence, the asymptotic theory of $U$-statistics is employed to arrive at the limiting null and alternative distributions of tests for all cases. These limit distributions are derived using a very mild and practically viable set of assumptions mainly on the traces of the unknown covariance matrices. Finally, corrections and improvements of a few other tests are also presented.

Place, publisher, year, edition, pages
2014. , 104 p.
Working paper / Department of Statistics, Uppsala University, 2014:4
Keyword [en]
Large dimensional matrices; Homogeneity, sphericity, identity; $U$-statistics; Non-normality;
National Category
Probability Theory and Statistics
Research subject
URN: urn:nbn:se:uu:diva-234031OAI: oai:DiVA.org:uu-234031DiVA: diva2:755099
Available from: 2014-10-13 Created: 2014-10-13 Last updated: 2014-10-15

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