In this paper the Rao score and likelihood ratio tests for hypothesis related to exchangeable structure of multivariate data covariance matrix are studied. Under the assumption of large-dimensionality the normal approximation of the Rao score test statistics distribution is proven as well as the exact and approximate distributions of the likelihood ratio test are derived. Simulation studies show the advantage of the Rao score test over the likelihood ratio test in both studied contexts: type I error and power. Moreover, the Rao score test is available in the case of high-dimensionality, and it is shown that the normal approximation matches well its distribution in this case. Thus, this latter approximation could be recommended for practical use.

Moments for the normal, Wishart and beta-type distributions are presented. A number of relations involving the trace function are also considered due to their connection to spectral moments of random matrices. For a couple of specific relations, the technique of obtaining the results is described in detail. The majority of the results are presented for real-valued matrices, but complex-valued matrices are occasionally also treated.

The authors derive the LRT statistic for the test of equality of mean vectors when the covariance matrix has what is called a double exchangeable structure. A second expression for this statistic, based on determinants of Wishart matrices with a block-diagonal parameter matrix, allowed for the expression of the distribution of this statistic as that of a product of independent Beta random variables. Moreover, the split of the LRT statistic into three independent components, induced by this second representation, will then allow for the expression of the exact distribution of the LRT statistic in a very manageable finite closed form for most cases and the obtention of very sharp near-exact distributions for the other cases. Numerical studies show that, as expected, due to the way they are built, these near-exact distributions are indeed asymptotic not only for increasing sample sizes but also for increasing values of all other parameters in the distribution, besides lying very close to the exact distribution even for extremely small samples.

Moments of functions of Wishart distributed matrices appear frequently in multivariate analysis. Although a considerable number of such moments have long been available in the literature, they appear in rather dispersed sources and may sometimes be difficult to locate. This paper presents a collection of moments of the Wishart and inverse Wishart distribution, involving functions such as traces, determinants, Kronecker, and Hadamard products, etc. Moments of factors resulting from decompositions of Wishart matrices are also included.

AbstractTraces of Wishart matrices appear in many applications, for example in finance, discriminant analysis, Mahalanobis distances and angles, loss functions and many more. These applications typically involve mixtures of traces of Wishart and inverse Wishart matrices that are concerned in this paper. Of particular interest are the sampling moments and their limiting joint distribution. The covariance matrix of the marginal positive and negative spectral moments is derived in closed form (covariance matrix of Y=[p?1Tr{W?1},p?1Tr{W},p?1Tr{W2}]?, where W?Wp(Σ=I,n)). The results are obtained through convenient recursive formulas for E[?i=0kTr{W?mi}] and E[Tr{W?mk}?i=0k?1Tr{Wmi}]. Moreover, we derive an explicit central limit theorem for the scaled vector Y, when p/n?d<1,p,n?∞, and present a simulation study on the convergence to normality and on a skewness measure.

The joint distribution of standardized traces of $\frac{1}{n}XX'$ and of $\Big(\frac{1}{n}XX'\Big)^2$, where the matrix $X:p\times n$ follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition $\frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of $n$ and $p$.

We derive an explicit formula for the R–transform of inverse Jacobi matrix I + W^−1 W2, where W1, W2 ∼ Wp(I, ni), i = 1, 2 are independent and I is p×p dimensional identity matrix using property of asymptotic freeness of Wishart and deterministic matrices. Procedure can be extended to other sets of the asymptotically free independent matrices. Calculations are illustrated with some simulations on fixed size matrices.

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.

Assume that a matrix X : p × n is matrix normally distributed and that the Kolmogorov condition, i.e., limn,p→∞ n = c > 0, holds. We propose a test for identity of the covariance matrix using a goodness-of-fit approach. Calculations are based on a recursive formula derived by Pielaszkiewicz et al. The test performs well regarding the power compared to presented alternatives, for both c < 1 or c ≥ 1. 

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 , which can be decomposed into the sum of asymptotically free independent summands.

Let  be a probability space. We consider the particular example of a non-commutative space, where  denotes the set of all   random matrices, with entries which are com-plex random variables with finite moments of any order and  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.

Finally, the main result is a new theorem pointing out classes of the matrix  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.

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).

In this paper, we give a general recursive formula for E(∏_i Tr{(WΣ^-1)^m_i}), where W~W_p(∑; n) denotes a real Wishart matrix. Formulas for xed n; p are presented as well as asymptotic versions when n/p→c, when n,p→∞ 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.

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.

The concepts of free cumulants and free moments are indispensably related to the idea of freeness introduced by Voiculescu [Voiculescu, D., Proc. Conf., Buşteni/Rom., Lect. Notes Math. 1132(1985), pp. 556-588] and studied further within Free probability theory. Free probability theory is of great importance for both the developing mathematical theories as well as for problem solving methods in engineering.

The goal of this paper is to present theoretical framework for free cumulants and moments, and then prove a new free cumulant-moment relation formula. The existing relations between these objects will be given. We consider as drawback that they require the combinatorial understanding of the idea of non--crossing partitions, which has been considered by Speicher [Speicher, R., Math. Ann., 298(1994), pp. 611-628] and then widely studied and developed by Speicher and Nica [Nica, A. and Speicher, R.:  Lectures on the Combinatorics of Free Probability, Cambridge University Press, Cambridge, United Kingdom, 2006]. 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, instead the calculations of the Stieltjes transform of the underlying measure are essential.

The presented free cumulant--moment relation formula is used to calculate cumulants of degree 1 to 5 as a function of the moments of lower degrees. The simplicity of the calculations can be observed by a comparison with the calculations performed in the classical way using non-crossing partitions. Then, the particular example of non-commutative space i.e., space of p×p matrices X=(X_ij)_ij, where X_ij has finite moments, equipped with functional E(TrX)∕p is investigated.

The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. In this thesis we combine the results of the theory together with the idea of free independence introduced by Voiculescu (1985).

Important theoretical part of the thesis consists of the introduction to Free probability theory, which justifies use of asymptotic freeness with respect to particular matrices as well as the use of Stieltjes and R-transform. Both transforms are presented together with their properties.

The aim of thesis is to point out characterizations of those classes of the matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider all matrices which can be decomposed to the sum of asymptotically free independent summands.

In particular, explicit calculations are performed in order to illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a matrix Q and generalize Marcenko and Pastur (1967) theorem. The matrix Q is defined as

where X_i is p × n matrix following a matrix normal distribution, X_i ~ N_p,n(0, \sigma^2I, I).

Finally, theorems pointing out classes of matrices Q which lead to closed formula for the asymptotic spectral distribution will be presented. Particularly, results for matrices with inverse Stieltjes transform, with respect to the composition, given by a ratio of polynomials of 1st and 2nd degree, are given.

The distribution of the vector of the normalized traces of  and of , where the matrix  follows a matrix normal distribution  and is proved, under the Kolmogorov condition , 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.