奇异线性模型的线性假设检验问题

本文研究奇异线性模型的假设检验问题，我们用初等直接的方法了根据最小二乘统一理论所构造的检验统计量服从Ｆ分布，并给出了这些结果在ｐａｎｅｌ数据模型。两级抽样回归模型以及不完全数据回归模型中的应用。     

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.     

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。     

贝叶斯LASSO正则加权复合分位回归及其应用（英文）

回归模型一般采取传统的最小二乘估计(LSE)方法,然而当数据包含非正态特征或异常值时该估计方法会导致不稳健的参数估计.与LSE方法相比,即使出现非正态误差或异常数据,复合分位回归(CQR)方法也能提供更稳健的估计结果.基于复合反对称拉普拉斯分布(CALD),本文提出了贝叶斯框架下的加权复合分量回归(WCQR)方法.正则化方法已经被验证可以有效处理高维稀疏回归模型,它可以同时进行变量选择和参数估计.本文结合贝叶斯LASSO正则化方法和WCQR方法来拟合线性回归模型,建立了 WCQR的贝叶斯LASSO正则化分层模型,并导出了所有参数的条件后验分布以进行统计推断.最后,通过蒙特卡罗模拟和实际数据分析演示了所提出方法.     

随机删失数据非线性回归模型的最小一乘估计

研究了随机删失数据非线性回归模型的最小一乘(LAD)估计问题, 证明了LAD估计量的渐近性质, 包括相合性、依概率有界性和渐近正态性等. 模拟结果显示对删失数据回归问题, LAD估计仍比最小二乘估计(LSE)稳健.     

生长曲线模型的分位数回归

生长曲线模型有着广泛的应用, 在经济学、生物学、医学等各个领域的研究都起着重要的作用. 已有文献关于生长曲线模型参数矩阵的估计基本上是使用最小二乘方法或极大似然方法. 使用最小二乘方法, 当误差项服从偏峰分布、厚尾分布、或者存在异常点时, 得出的估计不是有效的; 使用极大似然方法, 要求分布已知, 实际使用时很难满足这一点. 分位数回归能弥补如上这些缺陷, 所得估计具有很好的稳健性. 本文使用分位数回归方法给出生长曲线模型参数矩阵的估计, 及其渐近正态性.     

纵向数据均值和协方差矩阵的有效稳健估计（英文）

本文提出一种针对纵向数据回归模型下的均值和协方差矩阵同时进行的有效稳健估计.基于对协方差矩阵的Cholesky分解和对模型的改写,我们提出一个加权最小二乘估计,其中权重是通过广义经验似然方法估计出来的.所提估计的有效性得益于经验似然方法的优势,稳健性则是通过限制残差平方和的上界来达到.模拟研究表明,和已有的针对纵向数据的稳健估计相比,所提估计具有更高的效率和可比的稳健性.最后,我们把所提估计方法用来分析一组实际数据.     

Hermitian自反矩阵反问题的最小二乘解

本文研究了Hermitian自反矩阵反问题的最小二乘解及其最佳逼近.利用矩阵的奇异值分解理论,获得了最小二乘解的表达式.同时对于最小二乘解的解集合,得到了最佳逼近解.     

一种求解加权Toeplitz最小二乘问题的新预条件子（英文）

本文研究加权Toeplitz最小二乘问题的快速求解算法.首先,在增广线性系统的基础上,设计了一种用于求解此类线性系统的新型简单预条件子.其次,研究了迭代法的收敛性,并证明了预条件矩阵的所有特征值均是实数且非单位特征值位于某正区间.再次,研究了预条件矩阵的特征向量分布和最小多项式的维数.最后,相关数值实验表明新型预条件子比一些已有的预条件子更有效.     

Testing in linear composite quantile regression models

Composite quantile regression (CQR) can be more efficient and sometimes arbitrarily more efficient than least squares for non-normal random errors, and almost as efficient for normal random errors. Based on CQR, we propose a test method to deal with the testing problem of the parameter in the linear regression models. The critical values of the test statistic can be obtained by the random weighting method without estimating the nuisance parameters. A distinguished feature of the proposed method is that the approximation is valid even the null hypothesis is not true and power evaluation is possible under the local alternatives. Extensive simulations are reported, showing that the proposed method works well in practical settings. The proposed methods are also applied to a data set from a walking behavior survey.     

Derivative-free optimization and neural networks for robust regression

Large outliers break down linear and nonlinear regression models. Robust regression methods allow one to filter out the outliers when building a model. By replacing the traditional least squares criterion with the least trimmed squares (LTS) criterion, in which half of data is treated as potential outliers, one can fit accurate regression models to strongly contaminated data. High-breakdown methods have become very well established in linear regression, but have started being applied for non-linear regression only recently. In this work, we examine the problem of fitting artificial neural networks (ANNs) to contaminated data using LTS criterion. We introduce a penalized LTS criterion which prevents unnecessary removal of valid data. Training of ANNs leads to a challenging non-smooth global optimization problem. We compare the efficiency of several derivative-free optimization methods in solving it, and show that our approach identifies the outliers correctly when ANNs are used for nonlinear regression.     

Robust quaternion matrix completion with applications to image inpainting

In this paper, we study robust quaternion matrix completion and provide a rigorous analysis for provable estimation of quaternion matrix from a random subset of their corrupted entries. In order to generalize the results from real matrix completion to quaternion matrix completion, we derive some new formulas to handle noncommutativity of quaternions. We solve a convex optimization problem, which minimizes a nuclear norm of quaternion matrix that is a convex surrogate for the quaternion matrix rank, and the ?₁‐norm of sparse quaternion matrix entries. We show that, under incoherence conditions, a quaternion matrix can be recovered exactly with overwhelming probability, provided that its rank is sufficiently small and that the corrupted entries are sparsely located. The quaternion framework can be used to represent red, green, and blue channels of color images. The results of missing/noisy color image pixels as a robust quaternion matrix completion problem are given to show that the performance of the proposed approach is better than that of the testing methods, including image inpainting methods, the tensor‐based completion method, and the quaternion completion method using semidefinite programming.     

Regularization tools and robust optimization for ill-conditioned least squares problem: A computational comparison

Least squares problems arise frequently in many disciplines such as image restorations. In these areas, for the given least squares problem, usually the coefficient matrix is ill-conditioned. Thus if the problem data are available with certain error, then after solving least squares problem with classical approaches we might end up with a meaningless solution. Tikhonov regularization, is one of the most widely used approaches to deal with such situations. In this paper, first we briefly describe these approaches, then the robust optimization framework which includes the errors in problem data is presented. Finally, our computational experiments on several ill-conditioned standard test problems using the regularization tools, a Matlab package for least squares problem, and the robust optimization framework, show that the latter approach may be the right choice.     

Standard errors for the cox proportional hazards cure model

Standard errors for the maximum likelihood estimates of the regression parameters in the logistic-proportional-hazards cure model are proposed using an approximate profile likelihood approach and a nonparametric likelihood. Two methods are given and are compared with the standard errors obtained from the inverse of the joint observed information matrix of the regression parameters and the nuisance hazard parameters. The observed information matrix is derived and is shown to be an approximation of the conditional information matrix of the regression parameters given the hazard parameters. Simulations indicate that the standard errors obtained from the inverse of the observed information matrix based on the profile likelihood and the full likelihood are comparable and appropriate. The coverage rates for the logistic regression parameter are generally good. The proportional hazards regression parameter show reasonable coverage rates under ideal conditions but lower coverage rates when the incidence proportion is low or when censoring is heavy. The three methods are applied to a data set to investigate the effects of radiation therapy on tonsil cancer.     

多元测量误差模型的稳健GM-估计量

由于EV（Errores-in-Variables)模型（也称测量误差模型）的最大似然估计由正交回归给出，而正交回归对污染数据是敏感的，所以，需要采用稳健的统计方法来估计模型参数。本文在多元EV模型中引入稳健GM-估计量，把一元正态EV模型的若干结果推广到多元情形，所得的稳健性结果不仅更具一般性，而且还修正了文献中对一元情形给出的一个错误结果。     

锥优化模型在Robust桁架拓扑设计实例中的应用

本文讨论Robust桁架拓扑设计(TTD)问题，即桁架结构设计问题，使其在固定重量的情况下，具有最佳的承载能力．本文陈述了几种应用锥优化解Robust TTD问题的方法，并简介了锥优化最新的领域．同时，本文给出了一个单负荷的线性模型和一个多负荷的半正定优化模型以及Robust TTD问题．文中所有的模型均有例证．例证显示通过应用对偶性这些模型的规模能被充分的减小．     

一类分布鲁棒线性决策随机优化研究

随机优化广泛应用于经济、管理、工程和国防等领域,分布鲁棒优化作为解决分布信息模糊下的随机优化问题近年来成为学术界的研究热点.本文基于φ-散度不确定集和线性决策方式研究一类分布鲁棒随机优化的建模与计算,构建了易于计算实现的分布鲁棒随机优化的上界和下界问题.数值算例验证了模型分析的有效性.     

Robust reward–risk ratio optimization with application in allocation of generation asset

In this article, we study reward–risk ratio models under partially known message of random variables, which is called robust (worst-case) performance ratio problem. Based on the positive homogenous and concave/convex measures of reward and risk, respectively, the new robust ratio model is reduced equivalently to convex optimization problems with a min–max optimization framework. Under some specially partial distribution situation, the convex optimization problem is converted into simple framework involving the expectation reward measure and conditional value-at-risk measure. Compared with the existing reward–risk portfolio research, the proposed ratio model has two characteristics. First, the addressed problem combines with two different aspects. One is to consider an incomplete information case in real-life uncertainty. The other is to focus on the performance ratio optimization problem, which can realize the best balance between the reward and risk. Second, the complicated optimization model is transferred into a simple convex optimization problem by the optimal dual theorem. This indeed improves the usability of models. The generation asset allocation in power systems is presented to validate the new models.     

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In this article, we develop efficient robust method for estimation of mean and covariance simultaneously for longitudinal data in regression model. Based on Cholesky decomposition for the covariance matrix and rewriting the regression model, we propose a weighted least square estimator, in which the weights are estimated under generalized empirical likelihood framework. The proposed estimator obtains high efficiency from the close connection to empirical likelihood method, and achieves robustness by bounding the weighted sum of squared residuals. Simulation study shows that, compared to existing robust estimation methods for longitudinal data, the proposed estimator has relatively high efficiency and comparable robustness. In the end, the proposed method is used to analyse a real data set.