稳健矩阵回归模型和方法研究

随着大数据时代的到来,我们面临的数据越来越复杂,其中待估系数为矩阵的模型亟待构造和求解.无论在统计还是优化领域,许多专家学者都致力于矩阵模型的统计性质分析及寻找其最优解的算法设计.当随机误差期望为0且同方差时,采用基于最小二乘的模型可以很好地解决问题.当随机误差异方差,分布为重尾分布(如双指数分布,t-分布等)或数据含有异常值时,需要考虑稳健的方法来求解问题.常用的稳健方法有最小一乘,分位数,Huber等.目前稳健方法的研究大多集中于线性回归问题,对于矩阵回归问题的研究比较缺乏.本文从最小二乘模型讲起,对矩阵回归问题进行了总结和评述,同时列出了一些文献和简要介绍了我们的近期的部分工作.最后对于稳健矩阵回归,我们提出了一些展望和设想.     

用M方法检测回归系数矩阵的秩

为了确定多重线性回归模型中回归系数矩阵的秩，　本文提出了一个基于Ｍ估计的模型选择程序，　且在较弱的条件下建立了回归系数矩阵的秩的估计的强相合性。     

Rank estimation in reduced-rank regression

Reduced rank regression assumes that the coefficient matrix in a multivariate regression model is not of full rank. The unknown rank is traditionally estimated under the assumption of normal responses. We derive an asymptotic test for the rank that only requires the response vector have finite second moments. The test is extended to the nonconstant covariance case. Linear combinations of the components of the predictor vector that are estimated to be significant for modelling the responses are obtained.     

增长曲线模型中UMRE估计的存在性

对于设计矩阵不满秩，协方差阵任意或具有均匀结构或序列结构的正态增长曲线模型，本文讨论参数矩阵的一致最小风险同变（UMng）估计的存在性．在仿射变换群GI和转移交换群、二次损失和矩阵损失下本文分别获得存在回归系数矩阵的线性可估函数矩阵的UMRE估计的充要条件，推广了由[21]给出的在设计矩阵满秩下估计回归系数矩阵的结果．本文还首次证明了在群G1和二次损失下不存在协方差阵V和trV的UMRE估计．     

错误指定多元线性回归模型的Bayes估计

本文在错误指定下给出了多元线性模型的最优线性 Bayes估计 ,在矩阵损失下讨论了其相对于最小二乘法估计的优良性 ,且获得 Bayes估计的容许性和极小极大性     

线性模型回归系数EB估计的一致收敛速度

利用多元密度函数及其导数的核估计方法，建立了多元线性模型回归系数的经验Ｂａｙｅｓ估计，并给出了这种估计的一致收敛速度。     

带有变量中误差的 Probit 回归模型

一、引言描述一事件概率与影响变量间关系的 Probit 回归与 Logit 回归皆为重要的非线性模型.Logit 模型1944年由 Berkson 提出 Probit 模型1860年 Fechner 提出.对影响变量进行考察常会出现误差,称为“变量中的误差”(FV),在线性模型中的 FV问题已有系统研究,近来又扩展到各种非线性模型中.本文讨论 FV-Probit 模型.     

On estimation in some reduced rank extended growth curve models

The general multivariate analysis of variance model has been extensively studied in the statistical literature and successfully applied in many different fields for analyzing longitudinal data. In this article, we consider the extension of this model having two sets of regressors constituting a growth curve portion and a multivariate analysis of variance portion, respectively. Nowadays, the data collected in empirical studies have relatively complex structures though often demanding a parsimonious modeling. This can be achieved for example through imposing rank constraints on the regression coefficient matrices. The reduced rank regression structure also provides a theoretical interpretation in terms of latent variables. We derive likelihood based estimators for the mean parameters and covariance matrix in this type of models. A numerical example is provided to illustrate the obtained results.     

A Note on Random Effects Growth Curve Models

The robustness of regression coefficient estimator is a hot topic in regression analysis all the while. Since the response observations are not independent, it is extraordinarily difficult to study this problem for random effects growth curve models, especially when the design matrix is non-full of rank. The paper not only gives the necessary and sufficient conditions under which the generalized least square estimate is identical to the the best linear unbiased estimate when error covariance matrix is an arbitrary positive definite matrix, but also obtains a concise condition under which the generalized least square estimate is identical to the maximum likelihood estimate when the design matrix is full or non-full of rank respectively. In addition, by using of the obtained results, we get some corollaries for the the generalized least square estimate be equal to the maximum likelihood estimate under several common error covariance matrix assumptions. Illustrative examples for the case that the design matrix is full or non-full of rank are also given.     

Bayesian Analysis for Random Coefficient Regression Models Using Noninformative Priors

We apply Bayesian approach, through noninformative priors, to analyze a Random Coefficient Regression (RCR) model. The Fisher information matrix, the Jeffreys prior and reference priors are derived for this model. Then, we prove that the corresponding posteriors are proper when the number of full rank design matrices are greater than or equal to twice the number of regression coefficient parameters plus 1 and that the posterior means for all parameters exist if one more additional full rank design matrix is available. A hybrid Markov chain sampling scheme is developed for computing the Bayesian estimators for parameters of interest. A small-scale simulation study is conducted for comparing the performance of different noninformative priors. A real data example is also provided and the data are analyzed by a non-Bayesian method as well as Bayesian methods with noninformative priors.     

平衡损失下约束线性回归模型中回归系数的可容许估计

本文在平衡损失函数下得到等式约束模型中回归系数在齐次(非齐次)估计类中存在可容许估计的充要条件,给出带有不完全椭球约束模型中回归系数的线性估计在一切估计类中为可容许估计的充要条件.     

基于指数矩鞅差序列非参数回归函数的估计

本文研究了误差项是鞅差序列,且满足某种指数矩条件的非参数回归函数的估计.利用鞅的某种指数不等式,得到了其加权核估计的强相合以及在有限闭区间内一致强相合的性质,并在某种意义上推广了[5]的结果.     

增长曲线模型中UMRE估计和UMRU估计的存在性

对于一般的增长曲线模型,在一般的矩阵损失和二次损失下,用统一的方法分别给出了回归系数矩阵的任一指定可估函数存在一致最小风险同变（ＵＭＲＥ）估计（分别在仿真变换群和转换变换群下）和一致最小风险无编（ＵＭＲＵ）估计的充要条件,以及所有可估函数恒存在ＵＭＲＥ估计和ＵＭＲＵ估计的允要条件。最后将结果应用于一些特殊模型。     

Reduced-rank regression for the multivariate linear model

The problem of estimating the regression coefficient matrix having known (reduced) rank for the multivariate linear model when both sets of variates are jointly stochastic is discussed. We show that this problem is related to the problem of deciding how many principal components or pairs of canonical variates to use in any practical situation. Under the assumption of joint normality of the two sets of variates, we give the asymptotic (large-sample) distributions of the various estimated reduced-rank regression coefficient matrices that are of interest. Approximate confidence bounds on the elements of these matrices are then suggested using either the appropriate asymptotic expressions or the jackknife technique.     

半相依线性回归模型的影响分析

本文研究了半相依线性回归模型的影响分析问题.利用"数据删除"法,给出了基于(l)和E(l)的两种近似似然距离的具体计算公式,并且通过实际例子验证了本文的结论.     

多元协方差阵扰动模型岭估计的影响分析

本文研究了多元线性同归模型岭估计的影响分析问题.利用最小二乘估计方法,获得了多元协方差阵扰动模型与原模型参数阵之间的岭估计的一些关系式,给出了度量影响大小的基于岭估计的广义Cook距离.     

CONSERVATIVE ESTIMATING FUNCTIONIN THE NONLINEAR REGRESSION MODEL WITHAGGREGATED DATA

1 IntroductionWe consider following nonlinear regression model:where y = (yi, y25..., yn)" is an n x 1 response vector with expectationwhere pi(0) = f(xi, 0), i = 1, 2,... 5 n; 0 = (91, 92,... I or)" is a p x I vector of unknown parameters, 0 E O C Re; fi is a q x 1 desigll variable, fi E X, X C Rq, i = 1, 2,... t n; f(., .) is a knownfunction, its field of definition is X x O; V(P(0)) is a positive definite matrix for all 0 E O.Let Y be the n-dimensional sample space. Suppose that on …     

线性回归模型误差相关的一种诊断方法

本文就线性回归模型的优化问题 ,对误差的自相关性作了一些简单的分析 ,得出了一个对自相关现象的诊断方法 ,从而为模型的进一步诊断和治疗提供了依据和思路     

THE ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION IN MULTIPLE LINEAR REGRESSION MODEL

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误差为鞅差序列的回归函数估计的收敛速度

当误差为鞅差序列时,研究固定设计点列情形下非参数回归函数一般权函数的非参数估计,并在一些基本条件下给出了估计的一致最优强收敛速度．