本文選題：線性回歸模型 + NSD隨機序列��； 參考：《湖北師范大學》2017年碩士論文

【摘要】：線性回歸模型是當今統(tǒng)計模型中應用最基本最廣泛的一種模型,在工程技術、經(jīng)濟學和社會科學中具有很大的應用價值.然而,一些傳統(tǒng)的參數(shù)估計方法,如最小二乘法缺乏穩(wěn)健性.1964年Huber提出克服這一缺陷的M估計(參考文獻Huber[1]),此后,M估計便受到統(tǒng)計學者的廣泛關注與深入研究.很多回歸模型問題的研究中,常常假定誤差是獨立同分布的隨機變量.然而Huber指出,獨立誤差的條件比較保守,并且在很多實際應用中誤差常常表現(xiàn)出某種相依性,因此研究相依誤差的線性回歸模型具有很重要的理論與實際意義.負超可加相依(NSD)序列是一類較負相協(xié)(NA)序列更廣泛的序列,在介體理論和經(jīng)濟學中具有重要的應用價值.因此本文研究誤差為NSD線性回歸模型M估計的漸近性質(zhì),主要內(nèi)容如下:第二章,在合適的條件下,采用隨機變量的截斷方法建立了NSD隨機序列的中心極限定理和加權和中心極限定理.第三章,利用第二章中NSD隨機序列加權和的中心極限定理,證明了誤差為NSD的線性回歸模型(不含有常數(shù)項和含有常數(shù)項)M估計的漸近正態(tài)性.第四章,考慮第三章提到的誤差為NSD序列的回歸模型,基于M準則,提出了一個穩(wěn)健的M檢驗,并得到了M檢驗統(tǒng)計量的漸近分布.第五章,通過蒙特卡羅模擬方法,借用R軟件對回歸參數(shù)進行估計,并且計算出M檢驗的勢,證明了M檢驗統(tǒng)計量的漸近分布為2?分布.上述結論既推廣了Rao,Zhao[11]和Bai,Rao,Wu[12]對于誤差為獨立時線性回歸模型M估計漸近正態(tài)性的相應結論,又推廣了Zhao,Chen[13]關于線性回歸模型M檢驗統(tǒng)計量漸近分布的相應結論,同時也推廣了陳希孺和趙林城[8]中線性回歸模型M估計的漸近理論.
[Abstract]:Linear regression model is one of the most basic and widely used models in current statistical models. It has great application value in engineering, economics and social sciences. However, some traditional parameter estimation methods, such as the least square method, lack robustness. In 1964, Huber proposed M estimation to overcome this defect (Huber [1]). Since then, M estimation has been widely concerned and deeply studied by statisticians. In many studies of regression model problems, the error is often assumed to be a random variable with independent and same distribution. However, Huber points out that the condition of independent error is conservative, and the error often shows some dependence in many practical applications. Therefore, it is of great theoretical and practical significance to study the linear regression model of dependent error. The negative superadditive dependent (NSD) sequence is a more extensive class of sequences than the negative associative NAN sequence, which has important application value in mediator theory and economics. Therefore, in this paper, we study the asymptotic properties of M-estimators for NSD linear regression models. The main contents are as follows: in Chapter 2, under suitable conditions, The central limit theorem and weighted sum central limit theorem of NSD random sequence are established by using the truncation method of random variables. In chapter 3, by using the central limit theorem of the weighted sum of NSD random sequences in chapter 2, we prove the asymptotic normality of linear regression models with errors of NSD. In chapter 4, considering that the error mentioned in chapter 3 is the regression model of NSD sequence, a robust M-test is proposed based on M criterion, and the asymptotic distribution of M-test statistics is obtained. In chapter 5, by using Monte Carlo simulation method, the regression parameters are estimated by using R software, and the potential of M test is calculated, and the asymptotic distribution of M test statistics is proved to be 2? Distribution. The above conclusions not only generalize the corresponding conclusions of Rao Zhao [11] and Baihu Rao Wu [12] for the asymptotic normality of M estimation of linear regression model when the error is independent, but also generalize the corresponding conclusion of Zhaojian Chen [13] on the asymptotic distribution of M test statistics in linear regression model. The asymptotic theory of M-estimators for linear regression models in Chen Xiru and Zhao Lincheng [8] is also generalized.
【學位授予單位】：湖北師范大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：C815

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