發(fā)布時(shí)間：2018-04-30 15:13

  本文選題：正態(tài)線性模型 + 二次結(jié)構(gòu)的貝葉斯估計(jì)�。� 參考：《北京交通大學(xué)》2017年碩士論文

【摘要】：正態(tài)線性模型是線性模型中的一種,它在統(tǒng)計(jì)學(xué)中占有十分重要的地位。參數(shù)估計(jì)問(wèn)題是正態(tài)線性模型研究領(lǐng)域的核心問(wèn)題。常見(jiàn)的參數(shù)估計(jì)方法有最小二乘估計(jì)、極大似然估計(jì)和貝葉斯估計(jì)等,其中貝葉斯估計(jì)通常是樣本的非線性函數(shù),計(jì)算過(guò)程中涉及復(fù)雜的二重積分,難以獲得估計(jì)的顯式解。針對(duì)正態(tài)線性模型參數(shù)的估計(jì)問(wèn)題,本文提出了一種基于二次型統(tǒng)計(jì)量的貝葉斯估計(jì),這一方法結(jié)合了貝葉斯估計(jì)理論、樣本的充分統(tǒng)計(jì)量和二次型的相關(guān)知識(shí),既采納了先驗(yàn)信息,又避免了貝葉斯估計(jì)后驗(yàn)期望的復(fù)雜計(jì)算,在保證估計(jì)結(jié)果準(zhǔn)確性的同時(shí),提供了正態(tài)線性模型參數(shù)貝葉斯估計(jì)的顯式解。應(yīng)用這一方法,文章首先經(jīng)過(guò)推理計(jì)算,給出了正態(tài)線性模型參數(shù)基于三個(gè)統(tǒng)計(jì)量(?)、(?)2和(?)'X'X(?)的二次結(jié)構(gòu)的貝葉斯估計(jì)表達(dá)式。隨后,我們從理論上證明了在均方誤差矩陣準(zhǔn)則下,正態(tài)線性模型的參數(shù)基于三個(gè)統(tǒng)計(jì)量的二次結(jié)構(gòu)貝葉斯估計(jì)優(yōu)于基于兩個(gè)統(tǒng)計(jì)量(?)和(?)2的線性結(jié)構(gòu)貝葉斯估計(jì),并且優(yōu)于參數(shù)的極大似然估計(jì)和最小二乘估計(jì)。最后,分別基于參數(shù)的先驗(yàn)分布獨(dú)立與不獨(dú)立兩種情形,選取不同的先驗(yàn)分布,通過(guò)數(shù)值模擬考察二次結(jié)構(gòu)貝葉斯估計(jì)的優(yōu)良性,模擬結(jié)果顯示,隨著樣本容量n的增加和先驗(yàn)信息的逐漸集中,二次結(jié)構(gòu)貝葉斯估計(jì)越來(lái)越趨近于貝葉斯估計(jì)。
[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.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O212.8

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 ;The Superiorities of Bayes Linear Unbiased Estimator in Multivariate Linear Models[J];Acta Mathematicae Applicatae Sinica(English Series);2012年02期

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本文編號(hào)：1825062