正态线性模型参数的二次结构的贝叶斯估计

发布时间：2018-04-30 15:13

本文选题：正态线性模型 + 二次结构的贝叶斯估计　；参考：《北京交通大学》2017年硕士论文

【摘要】：正态线性模型是线性模型中的一种,它在统计学中占有十分重要的地位。参数估计问题是正态线性模型研究领域的核心问题。常见的参数估计方法有最小二乘估计、极大似然估计和贝叶斯估计等,其中贝叶斯估计通常是样本的非线性函数,计算过程中涉及复杂的二重积分,难以获得估计的显式解。针对正态线性模型参数的估计问题,本文提出了一种基于二次型统计量的贝叶斯估计,这一方法结合了贝叶斯估计理论、样本的充分统计量和二次型的相关知识,既采纳了先验信息,又避免了贝叶斯估计后验期望的复杂计算,在保证估计结果准确性的同时,提供了正态线性模型参数贝叶斯估计的显式解。应用这一方法,文章首先经过推理计算,给出了正态线性模型参数基于三个统计量(?)、(?)2和(?)'X'X(?)的二次结构的贝叶斯估计表达式。随后,我们从理论上证明了在均方误差矩阵准则下,正态线性模型的参数基于三个统计量的二次结构贝叶斯估计优于基于两个统计量(?)和(?)2的线性结构贝叶斯估计,并且优于参数的极大似然估计和最小二乘估计。最后,分别基于参数的先验分布独立与不独立两种情形,选取不同的先验分布,通过数值模拟考察二次结构贝叶斯估计的优良性,模拟结果显示,随着样本容量n的增加和先验信息的逐渐集中,二次结构贝叶斯估计越来越趋近于贝叶斯估计。
[Abstract]:Normal linear model is one of the linear models, which plays an important role in statistics. Parameter estimation is a core problem in the field of normal linear model. The commonly used parameter estimation methods include least square estimation, maximum likelihood estimation and Bayesian estimation. The Bayesian estimation is usually a nonlinear function of the sample, and complex double integrals are involved in the calculation process, so it is difficult to obtain the explicit solution of the estimation. In this paper, a Bayesian estimation based on quadratic statistics is proposed to estimate the parameters of normal linear model. This method combines Bayesian estimation theory, sufficient statistics of samples and knowledge of quadratic form. Not only the prior information is adopted, but also the complex computation of Bayesian estimation posteriori expectation is avoided. The explicit solution of Bayesian estimation of normal linear model parameters is provided while the accuracy of the estimation results is guaranteed. By using this method, the normal linear model parameters are given based on three statistical parameters, I. e. The Bayesian estimation expression of the quadratic structure of. Then, we prove theoretically that under the mean square error matrix criterion, the quadratic structure Bayesian estimation of normal linear model parameters based on three statistics is better than that based on two statistics. The linear structure Bayesian estimation is superior to the maximum likelihood estimation and least square estimation of the parameters. Finally, based on the independent and non-independent prior distribution of parameters, different prior distributions are selected, and the superiority of quadratic structure Bayesian estimation is investigated by numerical simulation. The simulation results show that, With the increase of sample size n and the gradual concentration of prior information, the quadratic structure Bayesian estimation approaches Bayesian estimation more and more.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.8

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