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Spectral analysis of the Moore-Penrose inverse of a large dimensional sample covariance matrix
Stockholm University, Faculty of Science, Department of Mathematics.
2016 (English)In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 148, 160-172 p.Article in journal (Refereed) Published
##### Abstract [en]

For a sample of $n$ independent identically distributed $p$-dimensional centered random vectorswith covariance matrix $\bSigma_n$ let $\tilde{\bS}_n$ denote the usual sample covariance(centered by the mean) and $\bS_n$ the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where $p> n$. In this paper, we provide the limiting spectral distribution andcentral limit theorem for linear spectralstatistics of the Moore-Penrose inverse of $\bS_n$ and $\tilde{\bS}_n$. We consider the large dimensional asymptotics when the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ such that $p/n\rightarrow c\in (1, +\infty)$. We present a Marchenko-Pastur law for both types of matrices, which shows that the limiting spectral distributions for both sample covariance matrices are the same. On the other hand, we demonstrate that the asymptotic distribution of linear spectral statistics of the Moore-Penrose inverse of $\tilde{\bS}_n$ differs in the mean from that of $\bS_n$.

##### Place, publisher, year, edition, pages
2016. Vol. 148, 160-172 p.
##### Keyword [en]
CLT, large-dimensional asymptotics, Moore-Penrose inverse, random matrix theory
##### National Category
Probability Theory and Statistics
Statistics
##### Identifiers
ISI: 000375826400012OAI: oai:DiVA.org:su-130903DiVA: diva2:934088
Available from: 2016-06-08 Created: 2016-06-08 Last updated: 2016-06-14Bibliographically approved

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Bodnar, Taras
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Department of Mathematics
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Journal of Multivariate Analysis
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Probability Theory and Statistics