| 1. |
- Liang, Yuli, 1985-
(författare)
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Contributions to Estimation and Testing Block Covariance Structures in Multivariate Normal Models
- 2015
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns inference problems in balanced random effects models with a so-called block circular Toeplitz covariance structure. This class of covariance structures describes the dependency of some specific multivariate two-level data when both compound symmetry and circular symmetry appear simultaneously.We derive two covariance structures under two different invariance restrictions. The obtained covariance structures reflect both circularity and exchangeability present in the data. In particular, estimation in the balanced random effects with block circular covariance matrices is considered. The spectral properties of such patterned covariance matrices are provided. Maximum likelihood estimation is performed through the spectral decomposition of the patterned covariance matrices. Existence of the explicit maximum likelihood estimators is discussed and sufficient conditions for obtaining explicit and unique estimators for the variance-covariance components are derived. Different restricted models are discussed and the corresponding maximum likelihood estimators are presented.This thesis also deals with hypothesis testing of block covariance structures, especially block circular Toeplitz covariance matrices. We consider both so-called external tests and internal tests. In the external tests, various hypotheses about testing block covariance structures, as well as mean structures, are considered, and the internal tests are concerned with testing specific covariance parameters given the block circular Toeplitz structure. Likelihood ratio tests are constructed, and the null distributions of the corresponding test statistics are derived.
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| 2. |
- Uwamariya, Denise, 1985-
(författare)
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Large deviations of condition numbers and extremal eigenvalues of random matrices
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Random matrix theory found applications in many areas, for instance in statistics random matrices are used to analyse multivariate data and their eigenvalues are used in hypothesis testing. Spectral properties of random matrices have been studied extensively in the literature dealing with both the bulk case (involving all the eigenvalues) and the extremal case (addressing the maximal and minimal eigenvalues). In this thesis two types of sequences of random matrices are considered: the first type is the sequence of sample covariance matrices, and the second type is the sequence of β-Laguerre (or Wishart) ensembles, for which large deviations of their extremal cases are studied. These two types of sequences of random matrices contain the classical Wishart matrices.The thesis can be divided into two parts. The first part is on the study of large deviations of condition numbers defined as ratios of maximal and the minimal eigenvalues. This is done based on suitable analysis and estimates of the joint density function of all eigenvalues. The second part deals with large deviations of individual maximal and minimal eigenvalue, and the approach consists of suitable eigenvalue concentration inequalities and Laplace’s method.It is remarked that for those two types of sequences of random matrices considered in this thesis, two scenarios are investigated: either one of the dimension size and the sample size is much larger than the other one, or the two sizes are comparable.
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| 3. |
- Byukusenge, Béatrice, 1984-
(författare)
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Residual Analysis in the GMANOVA-MANOVA Model
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis focuses on the establishment and analysis of residuals in the so called GMANOVA-MANOVA model. The model is a special case of the Extended Growth Curve Model. It has two terms where one term models the profiles (growth curves) and the other the covariables of interest. This model is useful in studying growth curves in short time series in fields such as economics, biology, medicine, and epidemiology. Furthermore, in the literature, residuals have been extensively studied and used to check model adequacy in univariate linear models. This thesis contributes to the extension of the study of residuals in the GMANOVA-MANOVA model. In this thesis, a new pair of residuals is established via the maximum likelihood estimators of the parameters in the model. One residual indicates whether an individual is far away from the group means and a second residual is used to check assumptions about the mean structure. Different properties of these residuals are verified and their interpretation is discussed. Moreover, using parametric bootstrap, the empirical distributions of the extreme elements in the residuals are derived. Finally, testing bilinear restriction in the MANOVA model is considered. One can show that the MANOVA model with bilinear restrictions is nothing more than a GMANOVA-MANOVA model. Furthermore, the likelihood ratio test can be shown to be given as a function of the residuals to the GMANOVA-MANOVA model, which can be used to understand the appropriateness of the model and test the bilinear hypothesis.
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| 4. |
- Ngailo, Edward Kanuti, 1982-
(författare)
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Contributions to linear discriminant analysis with applications to growth curves
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns contributions to linear discriminant analysis with applications to growth curves.Firstly, we present the linear discriminant function coefficients in a stochastic representation using random variables from the standard univariate distributions. We apply the characterized distribution in the classification function to approximate the classification error rate. The results are then extended to large dimension asymptotics under assumption that the dimension p of the parameter space increases together with the sample size n to infinity such that the ratio converges to a positive constant c (0, 1).Secondly, the thesis treats repeated measures data which correspond to multiple measurements that are taken on the same subject at different time points. We develop a linear classification function to classify an individual into one out of two populations on the basis of the repeated measures data that when the means follow a growth curve structure. The growth curve structure we first consider assumes that all treatments (groups) follows the same growth profile. However, this is not necessarily true in general and the problem is extended to linear classification where the means follow an extended growth curve structure, i.e., the treatments under the experimental design follow different growth profiles.At last, a function of the inverse Wishart matrix and a normal distribution finds its application in portfolio theory where the vector of optimal portfolio weights is proportional to the product of the inverse sample covariance matrix and a sample mean vector. Analytical expressions for higher order moments and non-central moments of the portfolio weights are derived when the returns are assumed to be independently multivariate normally distributed. Moreover, the expressions for the mean, variance, skewness and kurtosis of specific estimated weights are obtained. The results are complemented using a Monte Carlo simulation study, where data from the multivariate normal and t-distributions are discussed.
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| 5. |
- Ohlson, Martin, 1977-
(författare)
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Studies in Estimation of Patterned Covariance Matrices
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Many testing, estimation and confidence interval procedures discussed in the multivariate statistical literature are based on the assumption that the observation vectors are independent and normally distributed. The main reason for this is that often sets of multivariate observations are, at least approximately, normally distributed. Normally distributed data can be modeled entirely in terms of their means and variances/covariances. Estimating the mean and the covariance matrix is therefore a problem of great interest in statistics and it is of great significance to consider the correct statistical model. The estimator for the covariance matrix is important since inference on the mean parameters strongly depends on the estimated covariance matrix and the dispersion matrix for the estimator of the mean is a function of it.In this thesis the problem of estimating parameters for a matrix normal distribution with different patterned covariance matrices, i.e., different statistical models, is studied.A p-dimensional random vector is considered for a banded covariance structure reflecting m-dependence. A simple non-iterative estimation procedure is suggested which gives an explicit, unbiased and consistent estimator of the mean and an explicit and consistent estimator of the covariance matrix for arbitrary p and m.Estimation of parameters in the classical Growth Curve model when the covariance matrix has some specific linear structure is considered. In our examples maximum likelihood estimators can not be obtained explicitly and must rely on numerical optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linearly structured covariance matrix.This thesis also deals with the problem of estimating the Kronecker product structure. The sample observation matrix is assumed to follow a matrix normal distribution with a separable covariance matrix, in other words it can be written as a Kronecker product of two positive definite matrices. The proposed estimators are used to derive a likelihood ratio test for spatial independence. Two cases are considered, when the temporal covariance is known and when it is unknown. When the temporal covariance is known, the maximum likelihood estimates are computed and the asymptotic null distribution is given. In the case when the temporal covariance is unknown the maximum likelihood estimates of the parameters are found by an iterative alternating algorithm and the null distribution for the likelihood ratio statistic is discussed.
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| 6. |
- Umunoza Gasana, Emelyne, 1986-
(författare)
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An Edgeworth-type Expansion of the Distribution of a Likelihood-based Classifier for Single Time-point Measurements and Growth Curves
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis focuses on approximating misclassification errors of likelihood-based classifiers considering two cases. The first case assumes the allocation of a new observation into two normal populations. The second case classifies repeated measurements using the growth curve model, considering the fact that the new observation might not belong to any of the two predetermined populations but to an unknown population. In this thesis, likelihood-based approaches were used to derive classification rules used to allocate a new observation in any of the two predefined normally distributed populations. Moreover, a two-step likelihood-based classification of growth curves is studied from which the distribution of a new observation is either drawn from any of the two predetermined populations or from an unknown population. Furthermore, moments of the classifiers were calculated and utilized to approximate the distribution of the proposed classifiers through an Edgeworth-type expansion. In addition, probabilities of misclassifications for the above-mentioned classifiers were estimated.
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| 7. |
- Umunoza Gasana, Emelyne, 1986-
(författare)
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Misclassification Probabilities through Edgeworth-type Expansion for the Distribution of the Maximum Likelihood based Discriminant Function
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis covers misclassification probabilities via an Edgeworth-type expansion of the maximum likelihood based discriminant function. When deriving misclassification errors, first the expectation and variance in the population are assumed to be known where the variance is the same across populations and thereafter we consider the case where those parameters are unknown. Cumulants of the discriminant function for discriminating between two multivariate normal populations are derived. Approximate probabilities of the misclassification errors are established via an Edgeworth-type expansion using a standard normal distribution.
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| 8. |
- Erdtman, Elias, 1986-
(författare)
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Change point detection with respect to variance
- 2023
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis examines a simple method for detecting a change with respect to the variance in a sequence of independent normally distributed observations with a constant mean. The method filters out observations with extreme values and divides the sequence into equally large subsequences. For each subsequence, the count of extreme values is translated to a binomial random variable which is tested towards the expected number of extremes. The expected number of extremes comes from prior knowledge of the sequence and a specified probability of how common an extreme value should be. Then specifying the significance level of the goodness-of-fit test yields the number of extreme observations needed to detect a change. The approach is extended to a sequence of independent multivariate normally distributed observations by transforming the sequence to a univariate sequence with the help of the Mahalanobis distance. Thereafter it is possible to apply the same approach as when working with a univariate sequence. Given that a change has occurred, the distribution of the Mahalanobis distance of a multivariate normally distributed random vector with zero mean is shown to approximately follow a gamma distribution. The parameters for the approximated gamma distribution depend only on Σ1−1/2 Σ2Σ1−1/2 with Σ1 and Σ2 being the covariance matrices before and after the change has occurred. In addition to the proposed approach, other statistics such as the largest eigenvalue, the Kullback-Leibler divergence, and the Bhattacharyya distance are considered.
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| 9. |
- Lundengård, Karl, 1987-
(författare)
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Generalized Vandermonde matrices and determinants in electromagnetic compatibility
- 2017
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Matrices whose rows (or columns) consists of monomials of sequential powers are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility.In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, applications and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility.The second chapter focuses on finding the extreme points for the determinant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more.The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a standard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic discharge immunity as well as some experimentally measured data of similar phenomena.
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| 10. |
- Muhinyuza, Stanislas, 1980-
(författare)
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Statistical Inference of Tangency Portfolio in Small and Large Dimension
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis considers statistical test theory in portfolio theory. It analyses the asymptotic behavior of the considered tests in the high-dimensional setting, meaning k/n → c ∈ (0, ∞) as n → ∞, where k and n are portfolio size and sample size, respectively. It also considers the high-dimensional asymptotic of the product of components involved in the computation of the optimal portfolio. The thesis comprises four manuscripts:Paper I is concerned with the test on the location of the tangency portfolio on the set of feasible portfolios. Considering the independent and normally multivariate asset returns, we propose a finite-sample test on the mean-variance efficiency of the tangency portfolio (TP). We derive the distribution of the proposed test statistic under both the null and alternative hypotheses, using which we assess the power of the test and construct a confidence interval. The out-of-sample performance of the portfolio determined by the proposed test is conducted and through an extensive simulation study, we show the robustness of the developed test towards the violation of the normality assumptions. We also apply the developed test to real data in the empirical study.Paper II extends the results of paper I. It is concerned with the study of the asymptotic distributions of the test on the existence of efficient frontier (EF) and the efficiency of the tangency portfolio in the mean-variance space in the high-dimension setting under both the null and alternative hypotheses. Finite-sample performance and robustness of the proposed tests are studied through an extensive simulation study.In paper III, we study the distributional properties of the TP weights under the assumption of normally distributed logarithmic returns. The distribution of the weights of the TP is given under the form of a stochastic representation (SR). Using the derived SR we deliver the asymptotic distribution of the TP weights under a high-dimensional asymptotic regime. Besides, we consider tests about the elements of the TP weights and derive the asymptotic distribution of the test statistic under the null and alternative hypotheses. In a simulation study, we compare the power function of the high-dimensional asymptotic and the exact tests. Moreover, in an empirical study, we apply the developed theory in analysing the TP weights in a portfolio made of stocks from the S&P 500 index.In paper IV, we derive a stochastic representation of the product of a singular Wishart matrix and a singular Gaussian vector. We then use the derived SR in the obtention of the characteristic function of that product and in proving the asymptotic normality under the double asymptotic regime. The performance of the obtained asymptotic is shown in the simulation study.
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| 11. |
- Ngaruye, Innocent
(författare)
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Contributions to Small Area Estimation : Using Random Effects Growth Curve Model
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation considers Small Area Estimation with a main focus on estimation and prediction for repeated measures data. The demand of small area statistics is for both cross-sectional and repeated measures data. For instance, small area estimates for repeated measures data may be useful for public policy makers for different purposes such as funds allocation, new educational or health programs, etc, where decision makers might be interested in the trend of estimates for a specic characteristic of interest for a given category of the target population as a basis of their planning.It has been shown that the multivariate approach for model-based methods in small area estimation may achieve substantial improvement over the usual univariate approach. In this work, we consider repeated surveys taken on the same subjects at different time points. The population from which a sample has been drawn is partitioned into several non-overlapping subpopulations and within all subpopulations there is the same number of group units. The aim is to propose a model that borrows strength across small areas and over time with a particular interest of growth profiles over time. The model accounts for repeated surveys, group individuals and random effects variations.Firstly, a multivariate linear model for repeated measures data is formulated under small area estimation settings. The estimation of model parameters is discussed within a likelihood based approach, the prediction of random effects and the prediction of small area means across timepoints, per group units and for all time points are obtained. In particular, as an application of the proposed model, an empirical study is conducted to produce district level estimates of beans in Rwanda during agricultural seasons 2014 which comprise two varieties, bush beans and climbing beans.Secondly, the thesis develops the properties of the proposed estimators and discusses the computation of their first and second moments. Through a method based on parametric bootstrap, these moments are used to estimate the mean-squared errors for the predicted small area means. Finally, a particular case of incomplete multivariate repeated measures data that follow a monotonic sample pattern for small area estimation is studied. By using a conditional likelihood based approach, the estimators of model parameters are derived. The prediction of random effects and predicted small area means are also produced.
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| 12. |
- Ngaruye, Innocent
(författare)
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Small Area Estimation for Multivariate Repeated Measures Data
- 2014
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis considers Small Area Estimation with a main focus on estimation and prediction theory for repeated measures data. The demand for small area statistics is for both cross-sectional and repeated measures data. For instance, small area estimates for repeated measures data may be used by public policy makers for different purposes such as funds allocation, new educational or health programs and in some cases, they might be interested in a given group of population.It has been shown that the multivariate approach for model-based methods in small area estimation may achieve substantial improvement over the usual univariate approach. In this work, we consider repeated surveys including the same subjects at different time points. The population from which a sample has been drawn is partitioned into several subpopulations and within all subpopulations there is the same number of group units. For this setting a multivariate linear regression model is formulated. The aim of the proposed model is to borrow strength across small areas and over time with a particular interest of growth profiles over time. The model accounts for repeated surveys, group individuals and random effects variations.The estimation of model parameters is discussed with a restricted maximum likelihood based approach. The prediction of random effects and the prediction of small area means across time points, per group units and for all time points are derived. The theoretical results have also been supported by a simulation study and finally, suggestions for future research are presented.
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| 13. |
- Pielaszkiewicz, Jolanta Maria
(författare)
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Contributions to High–Dimensional Analysis under Kolmogorov Condition
- 2015
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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).
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| 14. |
- Pielaszkiewicz, Jolanta Maria, 1985-
(författare)
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On the asymptotic spectral distribution of random matrices : closed form solutions using free independence
- 2013
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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 Xi is p × n matrix following a matrix normal distribution, Xi ~ Np,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.
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| 15. |
- Uwamariya, Denise, 1985-
(författare)
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Large deviations of condition numbers of random matrices
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Random matrix theory has found many applications in various fields such as physics, statistics, number theory and so on. One important approach of studying random matrices is based on their spectral properties. In this thesis, we investigate the limiting behaviors of condition numbers of suitable random matrices in terms of large deviations.The thesis is divided into two parts. Part I, provides to the readers an short introduction on the theory of large deviations, some spectral properties of random matrices, and a summary of the results we derived, and in Part II, two papers are appended. In the first paper, we study the limiting behaviors of the 2-norm condition number of p x n random matrix in terms of large deviations for large n and p being fixed or p = p(n) → ∞ with p(n) = o(n). The entries of the random matrix are assumed to be i.i.d. whose distribution is quite general (namely sub- Gaussian distribution). When the entries are i.i.d. normal random variables, we even obtain an application in statistical inference. The second paper deals with the β-Laguerre (or Wishart) ensembles with a general parameter β > 0. There are three special cases β = 1, β = 2 and β = 4 which are called, separately, as real, complex and quaternion Wishart matrices. In the paper, large deviations of the condition number are achieved as n → ∞, while p is either fixed or p = p(n) → ∞ with p(n) = o(n/ln(n)).
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| 16. |
- Kuljus, Kristi, 1976-
(författare)
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Rank Estimation in Elliptical Models : Estimation of Structured Rank Covariance Matrices and Asymptotics for Heteroscedastic Linear Regression
- 2008
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis deals with univariate and multivariate rank methods in making statistical inference. It is assumed that the underlying distributions belong to the class of elliptical distributions. The class of elliptical distributions is an extension of the normal distribution and includes distributions with both lighter and heavier tails than the normal distribution.In the first part of the thesis the rank covariance matrices defined via the Oja median are considered. The Oja rank covariance matrix has two important properties: it is affine equivariant and it is proportional to the inverse of the regular covariance matrix. We employ these two properties to study the problem of estimating the rank covariance matrices when they have a certain structure.The second part, which is the main part of the thesis, is devoted to rank estimation in linear regression models with symmetric heteroscedastic errors. We are interested in asymptotic properties of rank estimates. Asymptotic uniform linearity of a linear rank statistic in the case of heteroscedastic variables is proved. The asymptotic uniform linearity property enables to study asymptotic behaviour of rank regression estimates and rank tests. Existing results are generalized and it is shown that the Jaeckel estimate is consistent and asymptotically normally distributed also for heteroscedastic symmetric errors.
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| 17. |
- Nzabanita, Joseph, 1977-
(författare)
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Bilinear and Trilinear Regression Models with Structured Covariance Matrices
- 2015
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis focuses on the problem of estimating parameters in bilinear and trilinear regression models in which random errors are normally distributed. In these models the covariance matrix has a Kronecker product structure and some factor matrices may be linearly structured. The interest of considering various structures for the covariance matrices in different statistical models is partly driven by the idea that altering the covariance structure of a parametric model alters the variances of the model’s estimated mean parameters.Firstly, the extended growth curve model with a linearly structured covariance matrix is considered. The main theme is to find explicit estimators for the mean and for the linearly structured covariance matrix. We show how to decompose the residual space, the orthogonal complement to the mean space, into appropriate orthogonal subspaces and how to derive explicit estimators of the covariance matrix from the sum of squared residuals obtained by projecting observations on those subspaces. Also an explicit estimator of the mean is derived and some properties of the proposed estimators are studied.Secondly, we study a bilinear regression model with matrix normally distributed random errors. For those models, the dispersion matrix follows a Kronecker product structure and it can be used, for example, to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimating equations, a flip-flop relation, are established.At last, the models based on normally distributed random third order tensors are studied. These models are useful in analyzing 3-dimensional data arrays. In some studies the analysis is done using the tensor normal model, where the focus is on the estimation of the variance-covariance matrix which has a Kronecker structure. Little attention is paid to the structure of the mean, however, there is a potential to improve the analysis by assuming a structured mean. We formally introduce a 2-fold growth curve model by assuming a trilinear structure for the mean in the tensor normal model and propose an estimation algorithm for parameters. Also some extensions are discussed.
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| 18. |
- Nzabanita, Joseph
(författare)
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Estimation in Multivariate Linear Models with Linearly Structured Covariance Matrices
- 2012
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis focuses on the problem of estimating parameters in multivariate linear models where particularly the mean has a bilinear structure and the covariance matrix has a linear structure. Most of techniques in statistical modeling rely on the assumption that data were generated from the normal distribution. Whereas real data may not be exactly normal, the normal distributions serve as a useful approximation to the true distribution. The modeling of normally distributed data relies heavily on the estimation of the mean and the covariance matrix. The interest of considering various structures for the covariance matrices in different statistical models is partly driven by the idea that altering the covariance structure of a parametric model alters the variances of the model’s estimated mean parameters.The extended growth curve model with two terms and a linearly structured covariance matrix is considered. In general there is no problem to estimate the covariance matrix when it is completely unknown. However, problems arise when one has to take into account that there exists a structure generated by a few number of parameters. An estimation procedure that handles linear structured covariance matrices is proposed. The idea is first to estimate the covariance matrix when it should be used to define an inner product in a regression space and thereafter reestimate it when it should be interpreted as a dispersion matrix. This idea is exploited by decomposing the residual space, the orthogonal complement to the design space, into three orthogonal subspaces. Studying residuals obtained from projections of observations on these subspaces yields explicit consistent estimators of the covariance matrix. An explicit consistent estimator of the mean is also proposed and numerical examples are given.The models based on normally distributed random matrix are also studied in this thesis. For these models, the dispersion matrix has the so called Kronecker product structure and they can be used for example to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimation equations in a flip-flop relation are presented and numerical examples are given.
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| 19. |
- Ohlson, Martin, 1977-
(författare)
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Testing spatial independence using a separable covariance matrix
- 2007
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Spatio-temporal processes like multivariate time series and stochastic processes occur in many applications, for example the observations from functional magnetic resonance imaging (fMRl) or positron emission tomography (PET). It is interesting to test independence between k sets of the variables, that is testing spatial independence.This thesis deals with the problem of testing spatial independence for dependent observations. The sample observation matrix is assumed to follow a matrix normal distribution with a separable covariance matrix, in other words it can be written as a Kronecker product of two positive definite matrices. Instead of having a sample observation matrix with independent columns, a covariance between the columns is considered and this covariance matrix is interpreted as a temporal covariance. The main results in this thesis are the computations of the maximum likelihood estimates and the null distribution of the likelihood ratio statistic. Two cases are considered, when the temporal covariance is known and when it is unknown. When the temporal covariance is known, the maximum likelihood estimates are computed and the asymptotic null distribution is shown to be similar to the independent observation case. In the case when the temporal covariance is unknown the maximum likelihood estimates of the parameters are found by an iterative alternating algorithm.A well known fact is that when testing hypotheses for covariance matrices, distributions of quadratic forms arise. A generalization of the distribution of the multivariate quadratic form X AX', where X is a (p x n) normally distributed matrix and A is a (n x n) symmetric real matrix, is presented. It is shown that the distribution of the quadratic form is the same as the distribution of a weighted sum of noncentral Wishart distributed matrices.
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