线性模型参数向量的近似贝叶斯估计  

Approximate Bayesian estimator for parameter vector in normal linear model

作　　者：蒋杰 王立春[1] JIANG Jie;WANG Li-chun(Dept.of Math.,Beijing Jiaotong Univ.,Beijing 100044,China)

机构地区：[1]北京交通大学理学院,北京100044

出　　处：《高校应用数学学报（A辑）》2022年第1期1-14,共14页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(11371051)。

摘　　要：用线性贝叶斯方法去同时估计线性模型中回归系数和误差方差,并在不知道先验分布具体形式的情况下,得到了线性贝叶斯估计的表达式.在均方误差矩阵准则下,证明了其优于最小二乘估计和极大似然估计.与利用MCMC算法得到的贝叶斯估计相比,线性贝叶斯估计具有显式表达式并且更方便使用.对于几种不同的先验分布,数值模拟结果表明线性贝叶斯估计比贝叶斯估计更接近真实值.进一步,通过模拟比较了线性贝叶斯估计和Lindley近似,从模拟结果可以发现线性贝叶斯估计有更好的估计效果.总之,不管是理论分析还是数值分析都表明了线性贝叶斯估计是一个有效可行的估计.In this paper, the linear Bayesian method is employed to estimate regression coefficient and error variance simultaneously in a linear model and the expression of linear Bayesian estimator(LBE) is obtained without specifying the specific form of the prior. It is proved that the LBE is superior to the ordinary least square estimator and the maximum likelihood estimator in terms of mean square error matrix criterion. The LBE not only has an explicit expression but is more convenient to use than the Bayesian estimator(BE) obtained by using MCMC algorithm. Numerical simulations show that the LBE is closer to the true value than the BE for several different priors. Further, compared with the Lindley approximation, the LBE has a better estimation effect. Hence, both theoretical analysis and numerical simulations show that the LBE is an effective and feasible estimator.

关 键 词：线性贝叶斯估计 均方误差矩阵 MCMC算法 Lindley近似 

分 类 号：O212.8[理学—概率论与数理统计]