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Contributions to High–Dimensional Analysis under Kolmogorov Condition
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
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 $\small\frac{p}{n}$ 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 $\small \frac{1}{p}E[Tr\{\cdot\}]$. 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 $\small E[\prod_{i=1}^k Tr\{W^{m_i}\}]$ 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 $\small \prod_{i=1}^k Tr\{W^{m_i}\}$, where $\small W\sim\mathcal{W}_p(I_p,n)$, is a consistent estimator of the $\small E[\prod_{i=1}^k Tr\{W^{m_i}\}]$. We consider

$\small Y_t=\sqrt{np}\big(\frac{1}{p}Tr\big\{\big(\frac{1}{n}W\big)^t\big\}-m^{(t)}_1 (n,p)\big),$,

where $\small m^{(t)}_1 (n,p)=E\big[\frac{1}{p}Tr\big\{\big(\frac{1}{n}W\big)^t\big\}\big]$, 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).

Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1724
Keyword [en]
Eigenvalue distribution; free moments; free Poisson law; Marchenko-Pastur law; random matrices; spectral distribution; Wishart matrix.
Mathematics
Identifiers
ISBN: 978-91-7685-899-8 (print)OAI: oai:DiVA.org:liu-122610DiVA: diva2:868746
Public defence
2015-12-11, Visionen, ingång 27, B-huset, Campus Valla, Linköping, 13:15 (English)
Supervisors
Available from: 2015-11-11 Created: 2015-11-11 Last updated: 2015-11-16Bibliographically approved
List of papers
1. Closed Form of the Asymptotic Spectral Distribution of Random Matrices Using Free Independence
Open this publication in new window or tab >>Closed Form of the Asymptotic Spectral Distribution of Random Matrices Using Free Independence
Abstract [en]

The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. Random matrix theory is the main eld placing its research interest in the diverse properties of matrices, with a particular emphasis placed on eigenvalue distribution. The aim of this article is to point out some classes of matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider matrices, later denoted by $\mathcal{Q}$, which can be decomposed into the sum of asymptotically free independent summands.

Let $(\Omega,\mathcal{F},P)$ be a probability space. We consider the particular example of a non-commutative space$(RM_p(\mathbb{C}),\tau)$, where $RM_p(\mathbb{C})$ denotes the set of all  $p \times p$ random matrices, with entries which are com-plex random variables with finite moments of any order and $\tau$ is tracial functional. In particular, explicit calculations are performed in order to generalize the theorem given in [15] and illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a particular form of matrix$Q\in\mathcal{Q}$.

Finally, the main result is a new theorem pointing out classes of the matrix $Q$ which leads to a closed formula for the asymptotic spectral distribution. Formulation of results for matrices with inverse Stieltjes transforms, with respect to the composition, given by a ratio of 1st and 2nd degree polynomials, is provided.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2015. 25 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2015:12
Keyword
closed form solutions, free probability, spectral distribution, asymptotic, random matrices, free independence.
Mathematics
Identifiers
urn:nbn:se:liu:diva-122170 (URN)LiTH-MAT-R--2015/12--SE (ISRN)
Available from: 2015-10-23 Created: 2015-10-23 Last updated: 2015-11-12Bibliographically approved
2. Cumulant-moment relation in free probability theory
Open this publication in new window or tab >>Cumulant-moment relation in free probability theory
2014 (English)In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 18, no 2, 265-278 p.Article in journal (Refereed) Published
Abstract [en]

The goal of this paper is to present and prove a cumulant-moment recurrent relation formula in free probability theory. It is convenient tool to determine underlying compactly supported distribution function. The existing recurrent relations between these objects require the combinatorial understanding of the idea of non-crossing partitions, which has been considered by Speicher and Nica. Furthermore, some formulations are given with additional use of the Möbius function. The recursive result derived in this paper does not require introducing any of those concepts. Similarly like the non-recursive formulation of Mottelson our formula demands only summing over partitions of the set. The proof of non-recurrent result is given with use of Lagrange inversion formula, while in our proof the calculations of the Stieltjes transform of the underlying measure are essential.

Place, publisher, year, edition, pages
University of Tartu Press, 2014
Keyword
R-transform, free cumulants, moments, free probability, non-commutative probability space, Stieltjes transform, random matrices
National Category
Probability Theory and Statistics Other Mathematics
Identifiers
urn:nbn:se:liu:diva-113087 (URN)10.12697/ACUTM.2014.18.22 (DOI)
Available from: 2015-01-08 Created: 2015-01-08 Last updated: 2017-12-05Bibliographically approved
3. On E [Pi(k)(i=0) Tr{W-mi}], where W similar to Wp (l, n)
Open this publication in new window or tab >>On E [Pi(k)(i=0) Tr{W-mi}], where W similar to Wp (l, n)
2017 (English)In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 46, no 6, 2990-3005 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we give a general recursive formula for $\small E[\prod_{i=0}^k Tr\{W^{m_i}\}]$, where $\small W \sim \mathscr{W}_p(I,n)$ denotes a real Wishart matrix. Formulas for fixed n, p  are presented as well as asymptotic versions when $\small \frac{n}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c$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, 2017
Keyword
Eigenvalue distribution; free moments; free Poisson law; Marchenko– Pastur law; random matrices; spectral distribution; Wishart matrix
Mathematics
Identifiers
urn:nbn:se:liu:diva-122618 (URN)10.1080/03610926.2015.1053942 (DOI)000390425800031 ()
Note

Available from: 2015-11-12 Created: 2015-11-12 Last updated: 2017-12-01Bibliographically approved
4. On p/n-asymptoticsapplied to traces of 1st and 2nd order powers of Wishart matrices with application to goodness-of-fit testing
Open this publication in new window or tab >>On p/n-asymptoticsapplied to traces of 1st and 2nd order powers of Wishart matrices with application to goodness-of-fit testing
Abstract [en]

The distribution of the vector of the normalized traces of $\small \frac{1}{n}XX'$ and of $\small \Big(\frac{1}{n}XX'\Big)^2$, where the matrix $\small X:p\times n$ follows a matrix normal distribution  and is proved, under the Kolmogorov condition $\small \frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$, to be multivariate normally distributed. Asymptotic moments and cumulants are obtained using a recursive formula derived in  Pielaszkiewicz et al. (2015). We use this result to test for identity of the covariance matrix using a goodness–of–fit approach. The test performs well regarding the power compared to presented alternatives, for both c < 1 or c ≥ 1.

Keyword
goodness–of–fit test, covariance matrix, Wishart matrix, multivariate normal distribution
Mathematics
Identifiers
urn:nbn:se:liu:diva-122620 (URN)
Available from: 2015-11-12 Created: 2015-11-12 Last updated: 2015-11-12Bibliographically approved

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Cite
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