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On E\big[\prod_{i=0}^k Tr\{W^{m_i}\} \big], where $W\sim\mathcal{W}_p(I,n)
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agricultural Sciences, Uppsala, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0001-9896-4438
2016 (English)In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415XArticle in journal (Refereed) Epub ahead of print
Abstract [en]

In this paper, we give a general recursive formula for , where  denotes a real Wishart matrix. Formulas for fixed n, p  are presented as well as asymptotic versions when i.e. when the so called Kolmogorov condition holds. Finally, we show  application of the asymptotic moment relation when deriving moments for the Marchenko-Pastur distribution (free Poisson law). A numerical  illustration using implementation of the main result is also performed.

Place, publisher, year, edition, pages
Taylor & Francis, 2016.
Keyword [en]
Eigenvalue distribution; free moments; free Poisson law; Marchenko– Pastur law; random matrices; spectral distribution; Wishart matrix
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-122618DOI: 10.1080/03610926.2015.1053942OAI: oai:DiVA.org:liu-122618DiVA: diva2:868929
Note

The previous status on this article was Manuscript.

Available from: 2015-11-12 Created: 2015-11-12 Last updated: 2016-04-13Bibliographically approved
In thesis
1. Contributions to High–Dimensional Analysis under Kolmogorov Condition
Open this publication in new window or tab >>Contributions to High–Dimensional Analysis under Kolmogorov Condition
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is about high–dimensional problems considered under the so{called Kolmogorov condition. Hence, we consider research questions related to random matrices with p rows (corresponding to the parameters) and n columns (corresponding to the sample size), where p > n, assuming that the ratio  converges when the number of parameters and the sample size increase.

We focus on the eigenvalue distribution of the considered matrices, since it is a well–known information–carrying object. The spectral distribution with compact support is fully characterized by its moments, i.e., by the normalized expectation of the trace of powers of the matrices. Moreover, such an expectation can be seen as a free moment in the non–commutative space of random matrices of size p x p equipped with the functional . Here, the connections with free probability theory arise. In the relation to that eld we investigate the closed form of the asymptotic spectral distribution for the sum of the quadratic forms. Moreover, we put a free cumulant–moment relation formula that is based on the summation over partitions of the number. This formula is an alternative to the free cumulant{moment relation given through non{crossing partitions ofthe set.

Furthermore, we investigate the normalized  and derive, using the dierentiation with respect to some symmetric matrix, a recursive formula for that expectation. That allows us to re–establish moments of the Marcenko–Pastur distribution, and hence the recursive relation for the Catalan numbers.

In this thesis we also prove that the , where , is a consistent estimator of the . We consider

,

where , which is proven to be normally distributed. Moreover, we propose, based on these random variables, a test for the identity of the covariance matrix using a goodness{of{t approach. The test performs very well regarding the power of the test compared to some presented alternatives for both the high–dimensional data (p > n) and the multivariate data (p ≤ n).

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. 61 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1724
Keyword
Eigenvalue distribution; free moments; free Poisson law; Marchenko-Pastur law; random matrices; spectral distribution; Wishart matrix.
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-122610 (URN)10.3384/diss.diva-122610 (DOI)978-91-7685-899-8 (print) (ISBN)
Public defence
2015-12-11, Visionen, ingång 27, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2015-11-11 Created: 2015-11-11 Last updated: 2015-11-16Bibliographically approved

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Pielaszkiewicz, Jolantavon Rosen, DietrichSingull, Martin
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