最小二乘估计关于误差分布的稳健性

对于设计矩阵$X$是列降秩的线性统计模型, 本文讨论了最小二乘估计关于误差分布的稳健性, 给出了误差分布的最大类, 使得误差项的分布在此范围内变动时, 最小二乘估计在均方误差矩阵准则下是最优估计.     

Gauss-Markov估计关于误差分布的稳健性

对于一般线性模型y=Xβ+ε,本文讨论了在广义均方误差准则及均方误差矩阵准则下,未知参数β的可估函数Xβ的Gauss-Markov估计关于误差分布的稳健性,分别给出了误差项ε的最大分布类,使得误差项ε的分布在此范围内变动时,Gauss-Markov估计在相应准则下是最优估计.     

线性模型中广义最小二乘参数估计误差分布的稳健性研究

研究了线性模型中广义最小二乘参数估计的误差分布稳健性问题.首先讨论了在线性统计模型里,设计矩阵为列降秩矩阵时,模型中给出了误差最大分布类,当误差向量的分布在此范围内变动时,LS估计和GLS估计在均方误差矩阵准则下是最优估计.然后进一步探讨广义最小二乘估计GLSE关于误差分布的稳健性,求出误差项所对应的最大分布族,进而证明了在该区间波动情况下,误差向量对应的始终为一致最优解.     

具有椭球等高分布误差的半参数回归模型参数的Bayes估计

本文给出了具有椭球等高分布误差的半参数回归模型中参数的Ｂａｙｅｓ估计．     

多维空间中变量含误差的直线模型

本文讨论了一种线性函数关系模型──多维空间中变量含误差的直线模型．利用一 些统计逼近定理及矩阵分析的结果,在关于真值及误差较弱的条件下证明了构造的参数估计 具有强相合性并且得到了ａ,ｓ收敛速度．在较强的条件下,还可得到估计量的渐近分布．     

部分线性度量误差模型中的经验似然推断

部分线性度量误差模型(Partial linear measurement error model)是经典的部分线性模型的推广．在此模型中,我们假定解释变量含有度量误差．本文,我们把经验似然推广到部分线性度量误差模型,得到了非参数的Wilk's定理．我们的方法可以用来构建置信区间(域),也可以用来检验．数值模拟表明,我们的方法在构建的置信区间长度以及覆盖率方面有很好的结果．     

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．     

回归模型的同方差检验

本文利用局部经验似然和WNW方法对条件分布函数和条件分位数进行估计,并利用条件分位数的方法对回归模型中的误差方差进行了同方差假设检验,获得了零假设下检验统计量的渐近分布为X2分布．模拟计算表明同方差假设检验的条件分位数方法具有较好的功效．     

核实数据下非线性半参数EV模型的经验似然推断

考虑带有协变量误差的非线性半参数模型,借助于核实数据,本文构造了未知参数的三种经验对数似然比统计量,证明了所提出的统计量具有渐近X2分布,此结果可以用来构造未知参数的置信域．另外,本文也构造了未知参数的最小二乘估计量,并证明了它的渐近性质．仅就置信域及其覆盖概率的大小方面,通过模拟研究比较了经验似然方法与最小二乘法的优劣．     

基于贝叶斯估计的中国人均GDP之时序模型

本文应用SAS软件对1952-2009年的中国人均GDP建立时间序列模型并对2010-2013年的中国人均GDP进行了预测;在此基础上建立了以时间序列模型得到的参数信息作为先验信息的两种贝叶斯修匀模型与算法。由此所得的参数贝叶斯估计及预测,能充分利用样本信息和参数的先验信息,因而具有更小的方差或平方误差,估计参数更科学。为了检验该方法对先验分布的灵敏性,我们做了基于两种先验分布的模拟预测。将预测结果与传统时间序列预测相比,发现单一正态观测值、方差已知的先验分布的贝叶斯模型得到的预测值更准确,而基于先验分布为指数分布的贝叶斯模型的预测误差较大,预测效果差。     

二类1错误矩阵方程增优置换变换求解及其在城市交通管理中的应用研究

消错学的错误矩阵可表达错误逻辑里所定义的分解、相似、增加、置换、毁灭、单位变换等转化词,针对其中的置换变换,构建了二类1错误矩阵方程增优置换变换错误矩阵方程,并讨论了该类错误矩阵方程的求解.用交通管理问题对错误矩阵进行了举例,并构建相应的错误矩阵方程,利用上述的求解方法,对二类1方程置换变换进行了求解.     

The role of data transfer on the selection of a single vs. multiple mesh architecture for tightly coupled multiphysics applications

Data transfer from one mesh to another may be necessary in a number of situations including spatial adaptation, remeshing, arbitrary Lagrangian-Eulerian (ALE), and multiphysics simulation. Data transfer has the potential to introduce error into a simulation; the magnitude and impact of which depends on the application, transfer scenario, and the algorithm used to perform the data transfer. During a transient simulation, data transfer may occur many times, with the potential of error accumulation at each transfer. This paper examines data transfer error and its impact on a set of simple multiphysics problems. Data transfer error is examined using analytical functions to compare schemes based on interpolation, area-weighted averaging, and L² minimization. An example error analysis is performed to illustrate data transfer error and behavior for a simple problem. Data transfer error is also investigated for a one-dimensional time-dependent system of partial differential equations. This study concludes that data transfer error can be significant in coupled multiphysics systems. These numerical experiments suggest that error is a function of data transfer scheme, and characteristics of the field data and mesh. If there are significant differences in the meshes in a multiple mesh simulation, this study suggests that data transfer may lead to error and instability if care is not taken. Further, this work motivates that data transfer error should be included in the estimation of numerical error when data transfer is employed in a simulation.     

带有变量选择过程的分类模型误差分析

偏倚一方差分析方法是在模型选择过程中权衡模型对现有样本解释程度和未知样本估计准确度的分析方法,目的是使选定的模型检验误差尽量小.在分类或回归过程中进行有效的变量筛选可以获得更准确的模型表达,但也会因此带来一定误差.提出"选择误差"的概念,用于刻画带有变量选择的分类问题中由于变量的某种选择方法所引起的误差.将分类问题的误差分解为偏倚—方差—选择误差进行研究,考察偏倚、方差和选择误差对分类问题的总误差所产生的影响.     

The limitations of nice mutually unbiased bases

Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show that the number of resulting mutually unbiased bases can be at most one plus the smallest prime power contained in the dimension, and therefore that this construction cannot improve upon previous approaches. We prove this by establishing a correspondence between nice mutually unbiased bases and abelian subgroups of the index group of a nice error basis and then bounding the number of such subgroups. This bound also has implications for the construction of certain combinatorial objects called nets.     

Forward error analysis of Gaussian elimination

Summary Part I of this work deals with the forward error analysis of Gaussian elimination for general linear algebraic systems. The error analysis is based on a linearization method which determines first order approximations of the absolute errors exactly. Superposition and cancellation of error effects, structure and sparsity of the coefficient matrices are completely taken into account by this method. The most important results of the paper are new condition numbers and associated optimal component-wise error and residual estimates for the solutions of linear algebraic systems under data perturbations and perturbations by rounding erros in the arithmetic floating-point operations. The estimates do not use vector or matrix norms. The relative data and rounding condition numbers as well as the associated backward and residual stability constants are scaling-invariant. The condition numbers can be computed approximately from the input data, the intermediate results, and the solution of the linear system. Numerical examples show that by these means realistic bounds of the errors and the residuals of approximate solutions can be obtained. Using the forward error analysis, also typical results of backward error analysis are deduced. Stability theorems and a priori error estimates for special classes of linear systems are proved in Part II of this work.     

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Summary. In this paper, we derive quasi-norm a priori and a posteriori error estimates for the Crouzeix-Raviart type finite element approximation of the p-Laplacian. Sharper a priori upper error bounds are obtained. For instance, for sufficiently regular solutions we prove optimal a priori error bounds on the discretization error in an energy norm when . We also show that the new a posteriori error estimates provide improved upper and lower bounds on the discretization error. For sufficiently regular solutions, the a posteriori error estimates are further shown to be equivalent on the discretization error in a quasi-norm. Received January 25, 1999 / Revised version received June 5, 2000 Published online March 20, 2001     

Robustness of the Hybrid Extragradient Proximal-Point Algorithm

The hybrid extragradient proximal-point method recently proposed by Solodov and Svaiter has the distinctive feature of allowing a relative error tolerance. We extend the error tolerance of this method, proving that it converges even if a summable error is added to the relative error. Furthermore, the extragradient step may be performed inexactly with a summable error. We present a convergence analysis, which encompasses other well-known variations of the proximal-point method, previously unrelated. We establish weak global convergence under mild assumptions.     

A Posteriori Error Estimators of Gradient Recovery Type for Elliptic Obstacle Problems

In this paper, we present a posteriori error estimates of gradient recovery type for elliptic obstacle problems. The a posteriori error estimates provide both lower and upper error bounds. It is shown to be equivalent to the discretization error in an energy type norm for general meshes. Furthermore, when the solution is smooth and the mesh is uniform, it is shown to be asymptotically exact. Some numerical results which demonstrate the theoretical results are also reported in this paper.     

Error bounds and estimates for eigenvalues of integral equations. II

Summary In a previous paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of non-symmetric integral equations. In this note an alternative analysis is presented leading to equivalent dominant error terms with error bounds which are quicker to calculate than those derived previously.     

A posteriori residual error estimation of a cell-centered finite volume method

We present an a posteriori residual error estimator for the Laplace equation using a cell-centered finite volume method in the plane. For that purpose we associate to the approximated solution a kind of Morley interpolant. The error is then the difference between the exact solution and this Morley interpolant. The residual error estimator is based on the jump of normal and tangential derivatives of the Morley interpolant. The equivalence between the discrete H¹-seminorm of the error and the residual error estimator is proved. The proof of the upper error bound uses the Helmholtz decomposition of the broken gradient of the error and some quasi-orthogonality relations. To cite this article: S. Nicaise, C. R. Acad. Sci. Paris, Ser. I 338 (2004).