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2020 | Book

Spectral Analysis of Large Reflexive Generalized Inverse and Moore-Penrose Inverse Matrices Read chapter Read first chapter

Recent Developments in Multivariate and Random Matrix Analysis

Festschrift in Honour of Dietrich von Rosen

Editors: Thomas Holgersson, Martin Singull

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume is a tribute to Professor Dietrich von Rosen on the occasion of his 65th birthday. It contains a collection of twenty original papers. The contents of the papers evolve around multivariate analysis and random matrices with topics such as high-dimensional analysis, goodness-of-fit measures, variable selection and information criteria, inference of covariance structures, the Wishart distribution and growth curve models.

Table of Contents

Frontmatter
Chapter 1. Spectral Analysis of Large Reflexive Generalized Inverse and Moore-Penrose Inverse Matrices
Abstract
A reflexive generalized inverse and the Moore-Penrose inverse are often confused in statistical literature but in fact they have completely different behaviour in case the population covariance matrix is not a multiple of identity. In this paper, we study the spectral properties of a reflexive generalized inverse and of the Moore-Penrose inverse of the sample covariance matrix. The obtained results are used to assess the difference in the asymptotic behaviour of their eigenvalues.
Taras Bodnar, Nestor Parolya
Chapter 2. Testing for Double Complete Symmetry
Abstract
In this paper the authors take an approach over the likelihood ratio test for double complete symmetry, which enables a particularly simple implementation of the test as well as a particularly adequate way towards obtaining very sharp approximations for the exact distribution of the test statistic. Numerical studies show the very good performance of the near-exact distributions derived, namely their very good performance for very small sample sizes and their asymptotic behavior for increasing numbers of variables involved.
Carlos A. Coelho, Martin Singull
Chapter 3. Convexity of Sets Under Normal Distribution in the Structural Alloy Steel Standard
Abstract
The paper is motivated by the structural alloy steel standard that has been used in China for a long period. This standard indicates the scope of several chemical elements in the steel and requests several mechanical properties for qualification. Fang and Wu (Acta Math Appl Sin 2:132–148, 1979) established the relationships between the percents of the controlled chemical elements and testing mechanical properties by a multivariate regression model, and proposed the algorithm for calculating qualification rate. Moreover, they proved the existence of the optimal chemical element combination. However, the uniqueness of the optimal solution for high dimensional case has been left. This open question is equivalent to showing the convexity of a type of probability sets under multivariate normal distribution. This paper proves that the open question is true.
Kai-Tai Fang, Zhen Luo, Yung Liang Tong
Chapter 4. Comments on Maximum Likelihood Estimation and Projections Under Multivariate Statistical Models
Abstract
Under the multivariate model with linearly structured covariance matrix with unknown variance components and known mean parameters (Szatrowski, Ann Stat 8:802–810, 1980) showed that the maximum likelihood estimators of variance components have explicit representation if and only if the space of covariance matrix is a quadratic subspace. The aim of this paper is to rewrite these results for models with unknown expectation and to give sufficient conditions for maximum likelihood estimator of covariance matrix to be a projection of the maximum likelihood estimator of unstructured covariance onto the space of structured matrices. The results will be illustrated by examples of structures suitable for multivariate models with general mean, growth curve models as well as doubly multivariate models.
Katarzyna Filipiak, Mateusz John, Augustyn Markiewicz
Chapter 5. Growth Curve Model with Orthogonal Covariance Structure
Abstract
In this paper we study the Growth Curve model with orthogonal covariance structure and derive estimators for all parameters. The orthogonal covariance structure is a generalization of many known structures, e.g., compound symmetry covariance structure. Hence, we compare our estimators with earlier results found in the literature.
Miguel Fonseca, Martin Singull
Chapter 6. Holonomic Gradient Method for the Cumulative Distribution Function of the Largest Eigenvalue of a Complex Wishart Matrix with Noncentrality Matrix of Rank One
Abstract
We apply the holonomic gradient method for evaluation of the cumulative distribution function of the largest eigenvalue of a complex Wishart matrix with the noncentrality matrix of rank one. This problem appears in the context of Rician fading of multiple-input/multiple-output (MIMO) wireless communications systems. We also give a brief survey of the use of multivariate analysis in wireless communication and the holonomic gradient method for statistical problems of performance evaluation in wireless communication.
Yuta Fukasawa, Akimichi Takemura
Chapter 7. Some Tests for the Extended Growth Curve Model and Applications in the Analysis of Clustered Longitudinal Data
Abstract
The Growth Curve Model (GCM) is a Generalized Multivariate Analysis of Variance (GMANOVA) model especially useful in the analysis of longitudinal data, growth curves as well as other response curves. The model is a natural extension of the classical Multivariate Analysis of Variance (MANOVA) model and among other assumptions, relies on the assumption that the mean for each group can be represented as a polynomial of degree q. The assumption that the mean over time for all groups follows a polynomial of the same degree is not always satisfied, since individuals across the different groups may respond differently. An excellent scenario is when we have clustered longitudinal data, where the response over time can be represented by polynomials of different degrees. In such situations, the natural extension is to use the Extended Growth Curve Model (EGCM), where one can assume different shapes to represent different groups or clusters. In this paper, we formulate hypotheses motivated by real life scenarios involving clustered longitudinal data, and propose tests that are motivated by residuals in the EGCM. We then mathematically derive the tests and evaluate performances using simulations. We provide real data examples as illustrations.
Jemila S. Hamid, Sayantee Jana
Chapter 8. Properties of BLUEs and BLUPs in Full vs. Small Linear Models with New Observations
Abstract
In this article we consider the partitioned linear model \( \mathcal {M}_{12} = \{ \mathbf {y}, \, {\mathbf {X}}_{1}\boldsymbol {\beta }_{1} + {\mathbf {X}}_{2}\boldsymbol {\beta }_{2}, \, \mathbf {V} \}\), where μ = X 1β 1 + X 2β 2, and the corresponding small model \( \mathcal {M}_{1} = \{ \mathbf {y}, \, {\mathbf {X}}_{1} \boldsymbol {\beta }_{1}, \, \mathbf {V} \}\), where μ 1 = X 1β 1. These models are supplemented with the new unobservable random vector y ∗, coming from y ∗ = Kβ 1 + ε ∗, where the covariance matrix of y ∗ is known as well as the cross-covariance matrix between y ∗ and y. We focus on comparing the BLUEs of μ 1 and μ, and BLUPs of y ∗ and ε ∗ under \( \mathcal {M}_{12} \) and \(\mathcal {M}_{1}\).
Stephen J. Haslett, Augustyn Markiewicz, Simo Puntanen
Chapter 9. A Collection of Moments of the Wishart Distribution
Abstract
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.
Thomas Holgersson, Jolanta Pielaszkiewicz
Chapter 10. Risk and Bias in Portfolio Optimization
Abstract
In this paper we derive weighted squared risk measures for a commonly used Stein-type estimator of the global minimum variance portfolio. The risk functions are conveniently split in terms of variance and squared bias over different weight matrices. It is argued that the common out-of-sample variance criteria should be used with care and that a simple unweighted risk function may be more appropriate.
Thomas Holgersson, Martin Singull
Chapter 11. Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit
Abstract
The chi-square distribution is often assumed to hold for the asymptotic distribution of two times the log likelihood ratio statistic under the null hypothesis. Approximations are derived for the mean and variance of G 2, the likelihood ratio statistic for testing goodness of fit in a s category multinomial distribution. The first two moments of G 2 are used to fit the distribution of G 2 to a noncentral chi-square distribution. The fit is generally better than earlier attempts to fit to scaled versions of asymptotic central chi-square distributions. The results enlighten the complex role of the dimension of the multivariate variable in relation to the sample size, for asymptotic likelihood ratio distribution results to hold.
Björn Holmquist, Anna Sjöström, Sultana Nasrin
Chapter 12. Covariance Structure Tests for t-distribution
Abstract
We derive expressions of statistics for testing covariance structures when the population is t-distributed. The likelihood ratio test, Rao’s score test and Wald’s score test are derived for basic covariance structures. Expressions of all three statistics are obtained under the general null-hypothesis H01 : Σ = Σ 0, using matrix derivative technique. Here p × p-matrix Σ is a dispersion/scale parameter. The special cases H02 : Σ = I p and H03 : Σ = γ 0I p where γ 0 > 0 is a known constant are also considered. Expressions of the statistics are obtained as approximations using first terms from Taylor expansions. The method can be carried over to other continuous multivariate elliptical distributions which have power function in the expression of the density function.
Tõnu Kollo, Marju Valge
Chapter 13. Variable Selection in Joint Mean and Covariance Models
Abstract
In this paper, we propose a penalized maximum likelihood method for variable selection in joint mean and covariance models for longitudinal data. Under certain regularity conditions, we establish the consistency and asymptotic normality of the penalized maximum likelihood estimators of parameters in the models. We further show that the proposed estimation method can correctly identify the true models, as if the true models would be known in advance. We also carry out real data analysis and simulation studies to assess the small sample performance of the new procedure, showing that the proposed variable selection method works satisfactorily.
Chaofeng Kou, Jianxin Pan
Chapter 14. On Shrinkage Estimators and “Effective Degrees of Freedom”
Abstract
Explicit expressions for the estimated mean \(\tilde {\mathbf {y}}_k = X\tilde {\boldsymbol {\beta }}_k = H_k\mathbf {y}\) and effective degrees of freedomν k = tr(H k) by penalized least squares, with penalty k||Dβ||2, can be found readily when X ′X + D ′D is nonsingular. We establish them here in general under only the condition that X be a non-zero matrix, and we show that the monotonicity properties that are known when X ′X is nonsingular also hold in general, but that they are affected by estimability of Dβ. We establish the relation between these penalized least squares estimators and least squares under the restriction that Dβ = 0.
Lynn R. LaMotte, Julia Volaufova, Simo Puntanen
Chapter 15. On Explicit Estimation of the Growth Curve Model with a Block Circular Covariance Structure
Abstract
Estimation of mean parameters in the growth curve model, when the covariance matrix has a block circular Toeplitz structure, is considered. The purpose of this article is to find the appropriate design matrices so that explicit mean estimators can be obtained.
Yuli Liang, Deliang Dai
Chapter 16. Space Decomposition and Estimation in Multivariate Linear Models
Abstract
Linear models are important in statistical analysis. In many real situations, these models become more and more complex, as such the estimation of model parameters constitutes a big challenge. To overcome this challenge many approaches have been proposed and space decomposition has emerged as a powerful tool in handling these complex models. This work gives a comprehensive review of some of challenges related to complex multivariate models and how the space decomposition has been successfully used. In this review, we first present the space decomposition as a tool to decompose complex models into tractable models that are easy to handle for estimation and testing. On the other hand, we give another space decomposition approach used for obtaining explicit estimators in multivariate linear models. Some examples on how this decomposition is performed for specific multivariate linear models are presented for both approaches.
Joseph Nzabanita, Innocent Ngaruye
Chapter 17. Detection of Sparse and Weak Effects in High-Dimensional Feature Space, with an Application to Microbiome Data Analysis
Abstract
We present a family of goodness-of-fit (GOF) test statistics specifically designed for detection of sparse-weak mixtures, where only a small fraction of the observational units are contaminated arising from a different distribution. The test statistics are constructed as sup-functionals of weighted empirical processes where the weight functions employed are the Chibisov-O’Reilly functions of a Brownian bridge. The study recovers and extends a number of previously known results on sparse detection using a weighted GOF (wGOF) approach. In particular, the results obtained demonstrate the advantage of our approach over a common approach that utilizes a family of regularly varying weight functions. We show that the Chibisov-O’Reilly family has important advantages over better known approaches as it allows for optimally adaptive, fully data-driven test procedures. The theory is further developed to demonstrate that the entire family is a flexible device that adapts to many interesting situations of modern scientific practice where the number of observations stays fixed or grows very slowly while the number of automatically measured features grows dramatically and only a small fraction of these features are useful. Numerical studies are performed to investigate the finite sample properties of the theoretical results. We shown that the Chibisov-O’Reilly family compares favorably to related test statistics over a broad range of sparsity and weakness regimes for the Gaussian and high-dimensional Dirichlet types of sparse mixture. Finally, an example of human gut microbiome data set is presented to illustrate that the family of tests has found applications in real-life sparse signal detection problems where the sample size is small in relation to the features dimension.
Tatjana Pavlenko, Annika Tillander, Justine Debelius, Fredrik Boulund
Chapter 18. Exploring Consistencies of Information Criterion and Test-Based Criterion for High-Dimensional Multivariate Regression Models Under Three Covariance Structures
Abstract
In this paper, we consider the high-dimensional consistency properties of an information criterion and a test-based criterion (KOO method) for the selection of variables in multivariate regression models with covariance structures. The covariance structures considered are (1) an independent covariance structure, (2) a uniform covariance structure and (3) an autoregressive covariance structure. In our model the sample size is not necessarily larger than the dimensionality (number) of response variables. Sufficient conditions for these criteria to be consistent are derived under a high-dimensional asymptotic framework such that the sample size and the dimensionality proceed to infinity together, with their ratio converging to a finite nonzero constant. Our results, and tendencies therein, are explored numerically through a Monte Carlo simulation.
Tetsuro Sakurai, Yasunori Fujikoshi
Chapter 19. Mean Value Test for Three-Level Multivariate Observations with Doubly Exchangeable Covariance Structure
Abstract
We consider matrix-valued multivariate observation model with three-level doubly-exchangeable covariance structure. We derive estimators of unknown parameters and their distributions under multivariate normality assumption. Test statistic for testing a mean value is proposed, and its exact distribution is derived. Several methods of computing p-values and critical values of the distribution are compared in real data example.
Ivan Žežula, Daniel Klein, Anuradha Roy
Chapter 20. Estimation of the Common Mean of Two Multivariate Normal Distributions Under Symmetrical and Asymmetrical Loss Functions
Abstract
In this paper, the estimation of the common mean vector of two multivariate normal populations is considered and a new class of unbiased estimators is proposed. Several dominance results under the quadratic loss and LINEX loss functions are established. To illustrate the usefulness of these estimators, a simulation study with finite samples is conducted to compare them with four existing estimators, including the sample mean and the Graybill-Deal estimator. Based on the comparison studies, we found that the numerical performance of the proposed estimators is almost as good as \(\tilde {\mu }_{CC}\) proposed by Chiou and Cohen (Ann Inst Stat Math 37:499–506, 1985) in terms of the risks. Its theoretical dominance over the sample mean of a single population under the sufficient conditions given is also established.
Dan Zhuang, S. Ejaz Ahmed, Shuangzhe Liu, Tiefeng Ma
Metadata
Title
Recent Developments in Multivariate and Random Matrix Analysis
Editors
Thomas Holgersson
Martin Singull
Copyright Year
2020
Electronic ISBN
978-3-030-56773-6
Print ISBN
978-3-030-56772-9
DOI
https://doi.org/10.1007/978-3-030-56773-6

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