含有可加和非可加测量误差的部分线性模型的统计推断
发布时间：2018-11-19 13:59
【摘要】：半参数回归模型在统计学理论中扮演着极其重要的角色,该模型为了充分利用数据中存在的相关信息,不仅引入参数分量还引入了非参数函数,极大的优化了模型的拟合效果.作为半参数模型的一种,部分线性模型结合了线性模型和非参数模型的特点,既具有线性模型容易解释的特点,又具有非参数模型的稳健性和灵活性,更重要的是它可以避免“维数灾祸”.这些特点使得该模型更加广泛的应用于实际问题.但是在解决实际问题时,我们通常不能直接获得变量的真实值,得到的数据是变量的观测值,这些观测值中往往存在各种误差,例如因为抽样引起的误差、因为实验过程中使用的工具或者是操作不规范引起的误差等等.我们称观测到的数据为真实协变量的代理变量,称协变量含有测量误差的问题为“测量误差问题”,称用于拟合带有测量误差数据的模型为“测量误差模型”在本文中,我们主要研究含有可加和非可加测量误差的半参数部分线性模型,其中参数部分协变量的观测值含有非可加测量误差,其真值需要借助辅助变量估计得到,非参数部分协变量的观测值含有可加测量误差.我们首先校正参数部分协变量,然后利用半参数最小二乘方法估计参数分量和非参数函数.在适当的假设条件下,证明了参数分量和非参数函数的渐近正态性、误差方差的渐近正态性,并进行了参数分量的广义似然比检验.最后,通过数值模拟与实际例子验证了文中所提出的估计方法以及检验方法.
[Abstract]:The semi-parametric regression model plays an important role in the statistical theory. In order to make full use of the relevant information in the data, the semi-parametric regression model not only introduces the parameter component but also the non-parametric function, which greatly optimizes the fitting effect of the model. As a semi-parametric model, the partial linear model combines the characteristics of the linear model and the non-parametric model, which has the characteristics of easy interpretation of the linear model, robustness and flexibility of the non-parametric model. More importantly, it can avoid the "dimensionality disaster". These characteristics make the model more widely used in practical problems. But when we solve practical problems, we usually can't get the real value of variables directly. The data we get are the observations of variables. There are often various errors in these observations, for example, because of the errors caused by sampling. Because of the tools used in the experiment or the error caused by the nonstandard operation, etc. We call the observed data the proxy variable of the real covariable, the problem that the covariable contains the measurement error as the "measurement error problem", and the model used to fit the data with the measurement error is called the "measurement error model" in this paper. We mainly study the semi-parametric partial linear model with additive and non-additive measurement errors, in which the observed values of some parameter covariables contain non-additive measurement errors, and the true values need to be obtained by the estimation of auxiliary variables. The observed values of nonparametric partial covariables contain additive measurement errors. We first calibrate some parameter covariables and then estimate parametric components and nonparametric functions by semi-parametric least square method. Under appropriate assumptions, the asymptotic normality of parametric components and nonparametric functions and the asymptotic normality of error variance are proved, and the generalized likelihood ratio test of parametric components is given. Finally, the proposed estimation method and the test method are verified by numerical simulation and practical examples.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O212.1
[Abstract]:The semi-parametric regression model plays an important role in the statistical theory. In order to make full use of the relevant information in the data, the semi-parametric regression model not only introduces the parameter component but also the non-parametric function, which greatly optimizes the fitting effect of the model. As a semi-parametric model, the partial linear model combines the characteristics of the linear model and the non-parametric model, which has the characteristics of easy interpretation of the linear model, robustness and flexibility of the non-parametric model. More importantly, it can avoid the "dimensionality disaster". These characteristics make the model more widely used in practical problems. But when we solve practical problems, we usually can't get the real value of variables directly. The data we get are the observations of variables. There are often various errors in these observations, for example, because of the errors caused by sampling. Because of the tools used in the experiment or the error caused by the nonstandard operation, etc. We call the observed data the proxy variable of the real covariable, the problem that the covariable contains the measurement error as the "measurement error problem", and the model used to fit the data with the measurement error is called the "measurement error model" in this paper. We mainly study the semi-parametric partial linear model with additive and non-additive measurement errors, in which the observed values of some parameter covariables contain non-additive measurement errors, and the true values need to be obtained by the estimation of auxiliary variables. The observed values of nonparametric partial covariables contain additive measurement errors. We first calibrate some parameter covariables and then estimate parametric components and nonparametric functions by semi-parametric least square method. Under appropriate assumptions, the asymptotic normality of parametric components and nonparametric functions and the asymptotic normality of error variance are proved, and the generalized likelihood ratio test of parametric components is given. Finally, the proposed estimation method and the test method are verified by numerical simulation and practical examples.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O212.1
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