浅谈广义逆矩阵

文章介绍莫尔-潘鲁斯(Moore-Penrose)广义逆矩阵的概念及其与实际背景的联系。文章中定理1和定理2说明条件I与相容线性方程组的基本解的广义逆矩阵的联系,定理3说明条件I和IV与相容线性方程组的最小模解的广义逆矩阵的联系。     

Necessary conditions for admissibility of matrix linear estimators in a multivariate linear model

This article provides necessary conditions for the admissibility of matrix linear estimators of an estimable parameter matrix linear function under two kinds of quadratic matrix loss functions in a multivariate linear model following a family of matrix normal distributions, where the covariance matrix associated is completely unknown. Further it is demonstrated that if a more concrete condition supplied for one of the subdivided conditions is satisfied, then the special condition concerning the Stein problem is necessary for the admissibility of the kind of estimators under each of the loss functions.     

Matlab软件在矩阵运算教学中的应用

文章以实例说明如何将Matlab软件融入到矩阵运算的教学中,解决矩阵计算中的冗繁性,增强学生应用数学软件和数学知识解决实际问题的能力,为线性代数的教学改革提供了一种新的模式。     

Decomposition of the Koopmans-Beckmann Problem

Subsequent findings indicate that the Koopmans-Beckmann linear programming representation of the quadratic assignment problem may be more useful in solving for the best integer assignment and a set of sustaining prices than their initial results suggest. In response to the renewed interest in the quadratic assignment problem, this brief paper presents the Koopmans-Beckmann linear program in general matrix form, demostrates that the program is decomposable, and shows that the primal subprogram constitutes a simple linear assignment problem whose optimal solution set always contains at least one integer assignment of plants to locations.     

The power of the Durbin-Watson test for regressions without an intercept

In the linear regression model without an intercept, the limiting power of the Durbin–Watson test (as correlation among errors increases) is shown to take only one of two values. This is either one or zero, depending on the underlying regressor matrix. Some examples and a simple rule to decide from a given regressor matrix which of these cases applies are also given.     

企业内部价值链优化模型研究——以西安市某装备制造企业为例

文章基于有效的管理企业价值活动这一目的,提出了一种层次分析法和模糊线性规划相结合的优化企业内部价值链的模型。模型中基于层次分析法推导出评价指标体系中各指标的综合权重;采用模糊线性规划法,建立适合价值链评价的综合数学模型,并通过测算得到最终的内部价值链优化模型。研究表明,层次分析法大大简化了各指标的权重计算,模糊线性规划使内部价值链管理的效果得到了定量研究,从而科学地建立了优化模型。     

The asymptotic behaviour of the estimated generalized least squares method in the linear regression model

Abstract  In the linear regression model the generalized least squares (GLS) method is only applicable if the covariance matrix of the errors is known but for a scalar factor. Otherwise an estimator for this matrix has to be used. Then we speak of the estimated generalized least squares (EGLS) method. In this paper the asymptotic behaviour of both methods is compared. Results are applied to some standard models commonly used in econometrics     

On equalities of estimations of parametric functions under a general linear model and its restricted models

Estimations of parametric functions under a general linear model and its restricted models involve some complicated operations of matrices and their generalized inverses. In the past several years, a powerful tool—the matrix rank method was utilized to manipulate various complicated matrix expressions that involve generalized inverses of matrices. In this paper, we use this method to derive necessary and sufficient conditions for six equalities of the ordinary least-squares estimators and the best linear unbiased estimators of parametric functions to equal under a general linear model and its corresponding restricted model.     

Optimal weighted average power similar tests for the covariance structure in the linear regression model

This paper suggests a procedure for the construction of optimal weighted average power similar tests for the error covariance matrix of a Gaussian linear regression model when the alternative model belongs to the exponential family. The paper uses a saddlepoint approximation to construct simple test statistics for a large class of problems and overcomes the computational burden of evaluating the complicated integrals arising in the derivation of optimal weighted average power tests. Extensions to panel data models are considered. Applications are given to tests for error autocorrelation in the linear regression model and in a panel data framework.     

Estimation strategies for the regression coefficient parameter matrix in multivariate multiple regression

We consider improved estimation strategies for the parameter matrix in multivariate multiple regression under a general and natural linear constraint. In the context of two competing models where one model includes all predictors and the other restricts variable coefficients to a candidate linear subspace based on prior information, there is a need of combining two estimation techniques in an optimal way. In this scenario, we suggest some shrinkage estimators for the targeted parameter matrix. Also, we examine the relative performances of the suggested estimators in the direction of the subspace and candidate subspace restricted type estimators. We develop a large sample theory for the estimators including derivation of asymptotic bias and asymptotic distributional risk of the suggested estimators. Furthermore, we conduct Monte Carlo simulation studies to appraise the relative performance of the suggested estimators with the classical estimators. The methods are also applied on a real data set for illustrative purposes.     

Simultaneous optimal estimation in linear mixed models

Simultaneous optimal estimation in linear mixed models is considered. A necessary and sufficient condition is presented for the least squares estimator of the fixed effects and the analysis of variance estimator of the variance components to be of uniformly minimum variance simultaneously in a general variance components model. That is, the matrix obtained by orthogonally projecting the covariance matrix onto the orthogonal complement space of the column space of the design matrix is symmetric, each eigenvalue of the matrix is a linear combinations of the variance components and the number of all distinct eigenvalues of the matrix is equal to the the number of the variance components. Under this condition, uniformly optimal unbiased tests and uniformly most accurate unbiased confidence intervals are constructed for the parameters of interest. A necessary and sufficient condition is also given for the equivalence of several common estimators of variance components. Two examples of their application are given.     

Approximation von entarteten linearen Modellen durch modelle mit vollem Rang

Metrika - In an application in the field of biochemistry we were asked whether the best linear unbiased estimator of the regression parameters in a linear model with degenerate covariance matrix...     

A note on least squares estimation and the blue in a generalized linear regression model

In a generalized linear regression model, least squares and Gauss-Markov estimators differ, in general, if the variance-covariance matrix of the disturbances is singular. In the present note it is shown that, nevertheless, the conventional least squares procedure leads to a Gauss-Markov estimator if it is applied to a modified model which results from adding dummy constraints to the original model. These constraints reflect the effects of the singularity of the variance- convariance matrix. As a consequence, a Gauss-Markov estimate may always be obtained by standard least squares minimization, which offers considerable computational advantages.     

More on connections between Wishart and matrix GIG distributions

The paper is devoted to relations between the matrix GIG and Wishart distributions. Our basic tool in the first part is a version of the Matsumoto-Yor property for matrix variables. This approach covers the following issues: the Herz identity for the Bessel function of matrix variate argument, characterization of a class of Wishart matrices and linear transformations of the matrix GIG distribution. The Bayesian Wishart model, studied in the second part, gives an alternative definition of the matrix GIG distribution. Such a model is characterized by linearity of conditional expectations and matrix GIG conditional distribution. It is also extended to Bayesian matrix GIG models, in the framework of which an interesting independence property is proved.     

不确定多时滞线性It随机系统的保性能控制

对一类具有多时滞的不确定线性随机系统,研究了保性能状态反馈控制律的设计问题。采用线性矩阵不等式方法和伊藤公式,导出了保性能控制律的存在条件。进而,通过设计一个线性无记忆的状态反馈控制器,使得对于所有系统容许的不确定,闭环系统是范数有界的。最后用数值例子说明了该方法的有效性。     

A test for the weights of the global minimum variance portfolio in an elliptical model

In this paper we consider the weights of the global minimum variance portfolio (GMVP). The returns are assumed to follow a matrix elliptically contoured distribution, i.e., the returns are assumed to be neither independent nor normally distributed. A test for the general linear hypothesis is given. The distribution of the test statistic is derived under the null and the alternative hypothesis. It turns out that its distribution is invariant with respect to the type of the matrix elliptical distribution, i.e., the stochastic properties of the GMVP do not depend either on the mean vector or on the distributional assumptions imposed on asset returns. In an empirical study we analyze an international diversified portfolio.     

The Trigonometry of Matrix Statistics

A matrix trigonometry developed chiefly by this author during the past 40 years has interesting applications to certain situations in statistics. The key conceptual entity in this matrix trigonometry is the matrix (maximal) turning angle. Associated entities (originally so-named by this author) are the matrix antieigenvalues and corresponding antieigenvectors upon which the matrix obtains its critical turning angles. Because this trigonometry is the natural one for linear operators and matrices, it also is the natural one for matrix statistics.     

解加权约束最小二乘问题的行M-不变方法

本文提出了一种新的求解加权约束线性最小二乘问题方法,即利用行M-不变矩阵得到了求解加权约束线性最小二乘的updating问题的递推方法。     

Statistical Proofs of Some Matrix Theorems *

Books on linear models and multivariate analysis generally include a chapter on matrix algebra, quite rightly so, as matrix results are used in the discussion of statistical methods in these areas. During recent years a number of papers have appeared where statistical results derived without the use of matrix theorems have been used to prove some matrix results which are used to generate other statistical results. This may have some pedagogical value. It is not, however, suggested that prior knowledge of matrix theory is not necessary for studying statistics. It is intended to show that a judicious use of statistical and matrix results might be of help in providing elegant proofs of problems both in statistics and matrix algebra and make the study of both the subjects somewhat interesting. Some basic notions of vector spaces and matrices are, however, necessary and these are outlined in the introduction to this paper.     

Approximate inference in multivariate nonlinear functional relationships

An estimation procedure based on estimating equations is presented for the parameters in a multivariate functional relationship model, where all observations are subject to error. The covariance matrix of the observational errors may be parametrized and is allowed to be different for different sets of observations. Estimators are defined for the unknown relation parameters and error parameters.
For linear models (i.e. where the model function is linear in the incidental parameters) the estimators are consistent and asymptotically normal. A consistent expression for the covariance matrix of the estimators is derived. The results are valid for general error distributions.
For nonlinear models the estimators are based on locally linear approximations to the model function. The afore mentioned properties of the estimators are now only approximately valid. The adequacy of the approximate inference, based on asymptotic theory for the linearized model, needs at least informal check. Some examples are given to illustrate the estimation procedure.