方差分量模型协方差阵的谱分解

本文对平衡方差分量模型, 给出了其协方差阵的新的谱分解算法. 该方法的特点是计算简单, 易于理解, 无须复杂的数学知识. 且能够明确显示协方差阵的不同特征值的个数, 以及谱分解中不同特征值所对应的投影阵的显式表示. 基于新方法我们进一步研究了平衡方差分量模型的一些相关性质.本文还研究了一般方差分量模型, 我们首先定义了一般方差分量模型协方差阵的简单谱分解,给出了一般方差分量模型可以进行简单谱分解的充要条件, 并研究了协方差阵简单谱分解的一些性质. 对于协方差阵可以进行简单谱分解的方差分量模型, 本文研究了简单谱分解在其统计推断中的应用.     

基于可导性方差分解的居民收入省际差距分析

本文提出了一个基于收入构成差异和收入差距动因的分解框架,旨在量化相关决定因素对居民人均收入省际差距的贡献度.从指标内在关联性维度将居民人均收入指标进行多指标分解,由此导出因变量指标与各自变量之和（或之乘积）之间存在恒等关系的表达式,并引入可导性方差分解法构造了地区间居民收入绝对差距和相对差距的结构与动因分解模型.研究结果表明,驱动2005-2012年中国居民人均收入省际间差距形成和缩小的首要动力是人均设备性资本,其次是非设备性资本与设备性资本比例;人力资本与总人口比例在差距形成和绝对差距缩小中具有显著的推动作用,但在相对差距缩小中表现出明显的抑制效应;非设备性资本产出率在差距形成中具有较大的推动作用,但在差距缩小中表现出巨大的抑制效应;劳动力与人力资本比例在差距形成和绝对差距缩小中具有显著的抑制作用,但在相对差距缩小中表现出巨大的推动作用;产出分配率在差距形成中发挥了较小的抑制作用,但在差距缩小中表现出巨大的遏制作用.     

多元线性回归置信域的局部影响

运用Ｃｏｏｋ（１９８６）的局部影响法评价多元线性回归模型的微小扰动对回归系数置信域的影响，扰动方式包括协方差阵扰动，自变量扰动和因变量扰动．     

多指标面板数据聚类方法及应用——以行业一次能源消费面板数据为例

综合考虑面板数据多指标中因变量指标特征及其与自变量指标的相关关系,通过定义因变量自协方差及自变量与因变量协方差以构建面板数据相似及相关性测度距离函数,在引入自组织竞争网络算法的基础之上,提出了多指标面板数据聚类的方法.以我国1996-2008年44个行业煤炭、天然气、电力消耗量及国际石油价格面板数据进行实际应用,验证了新方法聚类结果更为显著的优点。     

椭圆中的托勒密定理

著名的托勒密（Ptolemy）定理“圆内接四边形中,两条对角线的乘积等于两组对边乘积之和”有多种推广．但笔者未见在椭圆中的推广．其实,Ptolemy定理椭圆中也有．     

误差服从多元t分布的一类线性模型下参数估计的若干注记

研究一类线性模型下参数估计的若干问题．这类模型包含了多个因变量线性模型、增长曲线模型、扩充的增长曲线模型、似乎不相关回归方程组、方差分量模型等常用模型．在这类线性模型下,证明了当误差服从多元t分布时与误差服从多元正态分布时,具有相同的完全统计量和无偏估计,且在后一种情况下的充分统计量必为前一种情况下的充分统计量．对于带有多种协方差结构的前述几种模型,把在误差服从多元正态分布下,相应的协方差阵及有关参数的一致最小风险无偏(UMRU)估计存在性的结论推广到了相应的误差服从多元t分布情形．此外,对于误差服从多元t分布的这类统一的线性模型,给出了回归系数的线性可估函数的无偏估计的协方差阵的C-R下界．     

非正交试验的平方和分解及其应用

在方差分析中,平方和分解占有很重要的地位.在正交试验里,可以将平方和分解为各效应平方和和误差平方和之和,且使各效应平方和相互独立,从而对各效应假设分别作F 检验.但在实际中,我们经常碰到的数据并不是正交的,此时用最小二乘法得到的估计,其总平方和不再等于各效应平方和与误差平方和之和,而且计算复杂,应用起来极不     

含高维相依自变量的中心k阶条件矩子空间的估计

在回归分析中往往对条件均值,条件方差及高阶条件矩特别感兴趣.本文我们将关注中心k阶条件矩子空间在高维相依自变量情形的估计问题.为此,我们首先引入中心k阶条件矩子空间的概念,并研究该子空间的基本性质.针对高维相依自变量的复杂数据,为了避免预测变量协方差阵的逆矩阵的计算,本文提出用偏最小二乘方法来估计中心k阶条件矩子空间....     

随机变量二次型的协方差在混合效应模型中的应用

本文提出方差分量ANOVA估计的一种改进方法, 证明了对于一般的方差分量模型, 只要方差分量的ANOVA估计存在就可以通过此方法给出其改进形式, 并且在均方误差意义下优于ANOVA估计. 特别地, 对于单向分类随机效应模型, Kelly和Mathew^[1]对ANOVA估计的改进就是我们提出的改进方法的特殊形式, 这也给出了此类改进估计在均方误差意义下优于ANOVA估计的另一种合理的解释. 同时, 本文又将此思想应用到对谱分解估计的改进上. 本文应用协方差的简单性质证明了对带有一个随机效应的方差分量模型, 当随机效应的协方差阵只有一个非零特征值时, 随机效应方差分量谱分解估计在均方误差意义下总是优于ANOVA估计. 本文最后将第三节的结论推广到广义谱分解估计下, 同时给出广义谱分解估计待定系数的一个合理的取值.     

巧用托勒密定理解代数题

托勒密定理:圆内接四边形的两组对边乘积之和等于两条对角线的乘积,由于这个定理所揭示的是圆内接四边形的边与对角线的特定关系,因而在证明与圆有关的线段关系的几何命题中有着独特的作用,若     

The Spectral Decomposition of Covariance Matrices for the Variance Components Models

The aim of this paper is to propose a simple method to determine the number of distinct eigenvalues and the spectral decomposition of covariance matrix for a variance components model. The method introduced in this paper is based on a partial ordering of symmetric matrix and relation matrix. A method is also given for checking straightforwardly whether these distinct eigenvalues are linear dependent as functions of variance components. Some examples and applications to illustrate the results are presented.     

一个矩阵分解算法的推广及应用

对样本相关系数矩阵等行和分解算法作了简化和推广,使算法不仅可以应用在基于正态总体非独立样本的假设检验问题,也可以有效地运用在最优化算法中牛顿法等与二次函数极小化有关的问题上.     

Asymptotical behavior of the variance of sums of random variables with random replacements

In this paper, we introduce a scheme of summation of independent random variables with random replacements. We consider a series of double arrays of identically distributed random variables that are row-wise independent, but such that neighboring rows contain a random common part of the repeating terms. By this-scheme we bscribe a model of strongly dependent noise. To investigate the sample mean of this noise, we consider the sum of random variables over the whole double array and its conditional variance with respect to replacements. For columns of the arrays we prove a covariance inequality. As a corollary of it, we demonstrate the law of large numbers for conditional variances. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 124–143. Translated by A. Sudakov.     

The spectral decomposition of a covariance matrix for the balanced mixed analysis of variance model

We derive the spectral decomposition of a covariance matrix for the balanced mixed analysis of variance model. The derivation is based on determining the distinct eigenvalues of a covariance matrix and then obtaining a principal idempotent matrix for each distinct eigenvalue. Examples are given to illustrate the results.     

广义预测误差方差分解及其在证券市场冲击传递性研究中的应用

方差分解是向量自回归模型中研究各变量的冲击对所有内生变量预测误差贡献的方法。文章介绍了广义预测误差方差分解,同传统的正交预测误差方差分解相比,这种方法的特点是不受向量自回归模型中变量排序的影响。文章利用广义方差分解研究了沪市各个分类指数之间的关系,显示了系统冲击在各个行业指数之间传递的特点。     

A Complete Characterization of Multivariate Normal Stable Tweedie Models through a Monge-Ampère Property

Extending normal gamma and normal inverse Gaussian models, multivariate normal stable Tweedie (NST) models are composed by a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with the same variance equal to the fixed component. Within the framework of multivariate exponential families, the NST models are recently classified by their covariance matrices V(m) depending on the mean vector m. In this paper, we prove the characterization of all the NST models through their determinants of V(m), also called generalized variance functions, which are power of only one component of m. This result is established under the NST assumptions of Monge-Ampère property and steepness. It completes the two special cases of NST, namely normal Poisson and normal gamma models. As a matter of fact, it provides explicit solutions of particular Monge-Ampère equations in differential geometry.     

Weak signals in high‐dimensional regression: Detection,estimation and prediction

Regularization methods, including Lasso, group Lasso, and SCAD, typically focus on selecting variables with strong effects while ignoring weak signals. This may result in biased prediction, especially when weak signals outnumber strong signals. This paper aims to incorporate weak signals in variable selection, estimation, and prediction. We propose a two‐stage procedure, consisting of variable selection and postselection estimation. The variable selection stage involves a covariance‐insured screening for detecting weak signals, whereas the postselection estimation stage involves a shrinkage estimator for jointly estimating strong and weak signals selected from the first stage. We term the proposed method as the covariance‐insured screening‐based postselection shrinkage estimator. We establish asymptotic properties for the proposed method and show, via simulations, that incorporating weak signals can improve estimation and prediction performance. We apply the proposed method to predict the annual gross domestic product rates based on various socioeconomic indicators for 82 countries.     

On the exploration of linear latent effect for multivariate modeling

A data analysis method is proposed to derive a latent structure matrix from a sample covariance matrix. The matrix can be used to explore the linear latent effect between two sets of observed variables. Procedures with which to estimate a set of dependent variables from a set of explanatory variables by using latent structure matrix are also proposed. The proposed method can assist the researchers in improving the effectiveness of the SEM models by exploring the latent structure between two sets of variables. In addition, a structure residual matrix can also be derived as a by-product of the proposed method, with which researchers can conduct experimental procedures for variables combinations and selections to build various models for hypotheses testing. These capabilities of data analysis method can improve the effectiveness of traditional SEM methods in data property characterization and models hypotheses testing. Case studies are provided to demonstrate the procedure of deriving latent structure matrix step by step, and the latent structure estimation results are quite close to the results of PLS regression. A structure coefficient index is suggested to explore the relationships among various combinations of variables and their effects on the variance of the latent structure.