【摘要】：線性模型被廣泛的應(yīng)用在物理學(xué)、經(jīng)濟學(xué)、生物學(xué)、遺傳學(xué)、博弈論等領(lǐng)域,其含參估計模型及其相應(yīng)的有偏估計方法是使用數(shù)學(xué)求解實際問題這一領(lǐng)域研究的經(jīng)典課題之一。本文針對有偏估量中的嶺參數(shù)估計和M—估計兩類參數(shù)估量方式進行探討,并得到少許新的論斷。在對于有偏估計中的嶺估計章節(jié)中,對可以進行估量的線性函數(shù)的最小二乘估計(LSE)進行了介紹,并概況歸納了該方法的優(yōu)缺點。在這些基礎(chǔ)上,給出在滿足Gauss-Markov假設(shè)的模型下有偏估計嶺估計的概念及性質(zhì);探討了利用變換參數(shù)形式對嶺估計進行改進的方法;由于復(fù)共線性的存在,使用最小二乘估計方法經(jīng)常會造成估計參數(shù)在精度上的損失,引進相對效率的概念,以線性回歸模型為例,基于最小特征值和均方誤差,定義了兩種新的相對效率,討論了它們的上下界,并證明了在新定義的效率下,嶺估計的效率高于最小二乘估計的效率。在關(guān)于有偏估計中的M—估計的參數(shù)估計章節(jié)中,介紹了M—估計方法的相關(guān)性質(zhì)以及其國內(nèi)外研究現(xiàn)狀。在一個新的估計類中,結(jié)合嶺估計思想,改進了M—估計方法,探討了改進估計與M—估計的關(guān)系,同時,基于5個假設(shè),探究了其是否具有相合性以及收斂性,文章最后證明結(jié)合估計具有相合性與收斂性。
[Abstract]:Linear model is widely used in physics, economics, biology, genetics, game theory and so on. Its parametric estimation model and its corresponding biased estimation methods are one of the classical research topics in the field of solving practical problems with mathematics. In this paper, the methods of ridge parameter estimation and M- estimation in biased estimation are discussed, and some new conclusions are obtained. In the chapter of Ridge estimation in biased estimation, the least square estimation of linear function is introduced, and the advantages and disadvantages of this method are summarized. On the basis of these, the concept and properties of biased ridge estimation under the model satisfying Gauss-Markov hypothesis are given, and the method of improving ridge estimation by means of transformation parameter form is discussed, because of the existence of complex collinearity, Using the least square estimation method often results in the loss of the precision of the estimation parameters. The concept of relative efficiency is introduced. Taking the linear regression model as an example, two new relative efficiencies are defined based on the minimum eigenvalue and mean square error. In this paper, the upper and lower bounds are discussed, and it is proved that the efficiency of ridge estimation is higher than that of least square estimator under the new definition. In the chapter on parameter estimation of M- estimation in biased estimation, the related properties of M- estimation and its research status at home and abroad are introduced. In a new class of estimators, we improve the M- estimation method and discuss the relationship between the improved estimators and the M- estimators. At the same time, based on five assumptions, we discuss whether the M- estimators are consistent and convergent. Finally, it is proved that the combined estimation is consistent and convergent.
【學(xué)位授予單位】：青島科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O212

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