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.     

协方差阵估计的比较

本文在一般线性回归模型误差异方差情况下，通过计算机模拟对回归系数最小二乘估计的协方差矩阵的估计进行了比较。结果表明，当样本大小大于50时，回归系数的最小二乘估计具有较高的估计精度；其协方差矩阵的五种估计以普通最小二乘估计的协方差矩阵为最优。     

线性模型中部分根方估计的优良性

在均方误差矩阵(MSE-M)准则和在Pitman Closeness(PC)准则下,比较了部分根方估计相对于最小二乘估计的优良性.     

广义压缩最小二乘估计

本文引进了线性模型中回归系数的一个估计类。许多常用的估计,例如岭回归估计、主成分估计、压缩最小二乘估计以及迭代估计都属于这个估计类。本文讨论该估计类中估计的容许性问题以及矩阵均方误差准则下估计的比较问题。     

正态-逆Wishart先验下多元线性模型中经验Bayes估计的优良性

在正态-逆Wishart先验下研究了多元线性模型中参数的经验Bayes估计及其优良性问题.当先验分布中含有未知参数时,构造了回归系数矩阵和误差方差矩阵的经验Bayes估计,并在Bayes均方误差(简称BMSE)准则和Bayes均方误差阵(简称BMSEM)准则下,证明了经验Bayes估计优于最小二乘估计.最后,进行了Monte Carlo模拟研究,进一步验证了理论结果.     

NA样本半参数回归模型估计的矩相合性

考虑了误差为NA序列的半参数回归模型,利用非参数估计方法给出了模型参数的最小二乘估计和加权最小二乘估计,并在适当条件下得到了它们的矩相合性.     

岭估计优于最小二乘估计的条件

本文讨论在均方误差意义下岭估计优于最小二乘估计的问题，给出了岭估计优于小最小二乘估计的必要条件及较一般的充分条件。     

基于比例移动最小二乘近似的误差分析

相较于移动最小二乘近似方法,比例移动最小二乘近似法有效地克服了前者带来的矩阵病态这一问题,展示出了更好的数值稳定性和更高的计算精度.给出了比例移动最小二乘近似对函数及其任意阶导数的误差估计,并给出了数值算例来验证之前的理论分析结果,通过与移动最小二乘近似的比较,表明比例移动最小二乘近似能得到更快的收敛性和更稳定的计算性.     

一种有偏估计与最小二乘估计的两种新的相对效率

考察了线性回归模型的回归系数的一类有偏估计,在均方误差矩阵准则下将其与最小二乘估计(LSE)进行比较,导出了这类有偏估计相对于LSE的两种新的相对效率的上、下界.     

约束线性模型的条件部分根方估计

对于线性约束下的线性回归模型,针对设计矩阵的病态问题,提出一种条件部分根方估计.并在均方误差矩阵准则和Pitman Closeness准则下,比较了条件部分根方估计相对于约束最小二乘估计的优良性.     

The Superiorities of Simultaneous Empirical Bayes Estimation for the Regression Coefficients and Error-Variance in Linear Model

When the hyperparameters of prior distribution are partly known in linear model, the simultaneous parametric empirical Bayes estimators (PEBE) of the regression coefficients and error variance are constructed. The superiority of PEBE over the least squares estimator (LSE) of regression coefficients is investigated in terms of the the mean square error matrix (MSEM) criterion, and the superiority of PEBE over LSE of the error variance is discussed under the the mean square error (MSE) criterion. Finally, when all hyperparameters are unknown, the PEBE of regression coefficients and error variance are reconstructed and the superiority of them over LSE under the MSE criterion are studied by simulation methods.     

On asymptotic normality of sequential LS-estimate for unstable autoregressive process AR(2)

For estimating parameters in an unstable AR(2) model, the paper proposes a sequential least squares estimate with a special stopping time defined by the trace of the observed Fisher information matrix. It is shown that the sequential LSE is asymptotically normally distributed in the stability region and on its boundary in contrast to the usual LSE, having six different types of asymptotic distributions on the boundary depending on the values of the unknown parameters. The asymptotic behavior of the stopping time is studied.     

错误先验假定下回归系数Bayes估计的小样本性质

本在于错误指定的先验假定获得了回归系数的Ｂａｙｅｓ估计（ＢＥ）,并在均方误差矩阵准则下对其与最小二乘（ＬＳ）估计进行了比较,导出了它们的相对效率的界、讨论了在后验ＰｉｔｍａｎＣｌｏｓｅｎｅｓｓ准则下ＢＥ相对于ＬＳ估计的优良性。     

Accuracy and Stability of the Null Space Method for Solving the Equality Constrained Least Squares Problem

The null space method is a standard method for solving the linear least squares problem subject to equality constraints (the LSE problem). We show that three variants of the method, including one used in LAPACK that is based on the generalized QR factorization, are numerically stable. We derive two perturbation bounds for the LSE problem: one of standard form that is not attainable, and a bound that yields the condition number of the LSE problem to within a small constant factor. By combining the backward error analysis and perturbation bounds we derive an approximate forward error bound suitable for practical computation. Numerical experiments are given to illustrate the sharpness of this bound.     

Bayes估计的进一步研究及其Pitman最优性

对线性模型参数,讨论了Bayes估计的Pitman最优性,将已有结果进行了改进,去掉了附加条件,证明了在Pitman准则下,Bayes估计一致优于最小二乘估计(LSE),在此基础上,提出了一种基于先验信息的方差分量估计,通过和基于LSE的方差分量估计作比较,证明了新估计是无偏估计且有更小的均方误差．最后,证明了在Pitman准则下生长曲线模型参数的Bayes估计优于最佳线性无偏估计．     

几种最小二乘估计（LSE）效率的偏序

研究了几种最小二乘估计（ＬＳＥ）效率的偏序，证明了均方误差比（ＭＳＥＲ）效率在几种效率度量中是最优的。     

PARAMETER ESTIMATION FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SMALL STABLE NOISES FROM DISCRETE OBSERVATIONS

We study the least squares estimation of drift parameters for a class of stochastic differential equations driven by small α-stable noises, observed at n regularly spaced time points ti=i/n, i=1,···,n on [0, 1]. Under some regularity conditions, we obtain the consistency and the rate of convergence of the least squares estimator (LSE) when a small dispersion parameter ε→0 and n→∞ simultaneously. The asymptotic distribution of the LSE in our setting is shown to be stable, which is completely different from the classical cases where asymptotic distributions are normal.     

Superiority of empirical Bayes estimation of error variance in linear model

In this paper, the Bayes estimator of the error variance is derived in a linear regression model, and the parametric empirical Bayes estimator (PEBE) is constructed. The superiority of the PEBE over the least squares estimator (LSE) is investigated under the mean square error (MSE) criterion. Finally, some simulation results for the PEBE are obtained.     

增加试验数据对岭估计的均方误差的影响

在线性模型中,对于回归系数的岭估计和广义岭估计,本文给出了选择岭参数的条件,在此条件下,岭估计和广义岭估计不仅能改善LS估计,而且增加试验数据时,它们的均方误差都会减少。同时,本文将增加试验数据换成增加附加信息,从而讨论了附加信息对混合岭估计和混合广义岭估计的影响问题。     

The superiorities of bayes linear unbiased estimator in multivariate linear models

In this article,the Bayes linear unbiased estimator (BALUE) of parameters is derived for the multivariate linear models.The superiorities of the BALUE over the least square estimator (LSE) is studied in terms of the mean square error matrix (MSEM) criterion and Bayesian Pitman closeness (PC) criterion.