部分线性变系数模型的岭估计

发布时间：2018-04-16 23:28

本文选题：部分线性变系数模型 + 轮廓最小二乘估计　；参考：《中央民族大学》2016年硕士论文

【摘要】：为了更好地拟合数据,人们提出了许多不同形式的非参数回归模型,其中的变系数回归模型既保持了非参数回归模型的灵活性,又可以很好地克服维数灾难问题,而部分线性变系数模型作为变系数模型的推广,反映了回归函数更为精确的结构,为分析和了解自变量对因变量的影响提供了更详细的信息,得到了广泛的应用。关于部分线性变系数模型,其参数分量的研究是重点。然而,目前的估计方法,不论是基于何种光滑技术,大都假设了自变量,特别是参数部分的自变量之间不存在复共线性。然而,实际数据分析中,复共线性是常见的问题,因此如何克服复共线性是回归分析的重要组成部分。众所周知,对于普通的线性回归模型来说,构造有偏估计,以牺牲无偏性来换取较小的均方误差是一个可行的途径。岭估计和主成分估计是教科书上最常提到的两种有偏估计。然而,绝大多数有偏估计和复共线性的讨论都是基于普通的线性回归模型,对于部分线性变系数模型来说,如何克服复共线性和构造对应的有偏估计是非常有意义的课题。但是,目前相关结果非常少。从实际应用和理论研究两个角度来说,研究部分线性变系数模型如何克服复共线性问题,如何构造有偏估计并研究其性质,这些研究都是必要和有意义的。本文针对部分线性变系数模型参数部分出现复共线性问题,主要做了以下工作：文章第二部分构造了部分线性变系数模型的轮廓最小二乘岭估计,并在均方误差意义下证明了在满足一定条件下,部分线性变系数模型的轮廓最小二乘岭估计是优于轮廓最小二乘估计的,并对模型进行了数值模拟；文章第三部分在考虑参数存在线性等式约束的条件下,构造了部分线性变系数模型的约束轮廓最小二乘岭估计,在均方误差意义下证明了存在岭参数使得约束轮廓最小二乘岭估计优于约束轮廓最小二乘估计,并对模型的估计做了数值模拟；文章第四部分对部分线性变系数模型的参数部分附加线性随机约束,构造了部分线性变系数模型的随机约束轮廓最小二乘岭估计,讨论了随机约束轮廓最小二乘岭估计的一些性质,并用数值模拟在均方误差下比较了随机约束轮廓最小二乘岭估计和存在随机约束的轮廓最小二乘估计。
[Abstract]:In order to fit the data better, many non-parametric regression models have been proposed. The variable coefficient regression model not only keeps the flexibility of the non-parametric regression model, but also can overcome the problem of dimensionality disaster.The partial linear variable coefficient model, as a generalization of the variable coefficient model, reflects the more accurate structure of the regression function and provides more detailed information for the analysis and understanding of the influence of independent variables on dependent variables, and is widely used.The research on the parameter components of partial linear variable coefficient model is the key point.However, most of the current estimation methods, regardless of the smoothing technique, assume that there is no complex collinearity between the independent variables, especially those in the parametric part.However, in practical data analysis, complex collinearity is a common problem, so how to overcome complex collinearity is an important part of regression analysis.It is well known that for ordinary linear regression models, it is a feasible way to construct biased estimators and exchange unbiased errors for smaller mean square errors.Ridge estimation and principal component estimation are the two most commonly mentioned biased estimates in textbooks.However, most of the discussions of biased estimation and complex collinearity are based on ordinary linear regression models. For partial linear variable coefficient models, how to overcome complex collinearity and construct corresponding biased estimators is a very meaningful topic.However, there are very few results so far.From the point of view of practical application and theoretical study, it is necessary and meaningful to study how partial linear variable coefficient model can overcome the problem of complex collinearity, how to construct biased estimation and study its properties.The main work of this paper is as follows: in the second part of this paper, we construct the contour least-squares estimation of the partial linear variable coefficient model.In the sense of mean square error, it is proved that the contour least square ridge estimation of partial linear variable coefficient model is superior to that of contour least square estimation under certain conditions, and the model is simulated numerically.In the third part of the paper, the constrained contour least square ridge estimation of partial linear variable coefficient model is constructed under the condition that the parameters have linear equality constraints.In the sense of mean square error, it is proved that the constrained contour least square ridge estimation is superior to the constrained contour least square estimation due to the existence of ridge parameters, and the model estimation is simulated numerically.In the fourth part of the paper, the parameters of the partially linear variable coefficient model are partially subject to linear random constraints, and the stochastic constrained contour least square ridge estimation of the partially linear variable coefficient model is constructed.In this paper, some properties of stochastic constrained contour least square ridge estimation are discussed, and the stochastic constrained contour least square estimation and the contour least square estimation with random constraints are compared by numerical simulation under mean square error.
【学位授予单位】：中央民族大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O212.1

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