带有测量误差的部分非线性变系数模型的统计推断

发布时间：2018-05-24 05:29

本文选题：测量误差 + 纠偏轮廓最小二乘　；参考：《南京理工大学》2017年硕士论文

【摘要】：回归模型是统计学中占有重要地位的模型之一,它的理论研究丰富,应用较为广泛。而作为回归模型中一个重要的分支,半参数模型以其结合参数模型和非参数模型的特点而受到广泛关注。该模型结合参数分量和非参数函数,充分挖掘数据之间的信息,使模型的拟合效果达到最大化。部分非线性变系数模型作为半参数模型之一,它继承了半参数模型的特点,不但消减了模型偏差,而且避免了"维数祸根"。这些特点使得该模型更能应用到实际问题中。但在解决实际问题时,往往有各种原因使得数据不能精确观测。例如,测量过程中使用的工具或操作不规范、外界环境的影响、抽样等。在统计研究中,我们通常把带有测量误差的问题称为"测量误差问题",把带有测量误差的模型称为"测量误差模型"。本文,我们主要研究带有测量误差的部分非线性变系数模型,其中非参数部分的协变量带有误差,并把误差分为可加和不可加两种情况。对带有可加测量误差的模型,我们采用纠偏的轮廓最小二乘分别得到参数与非参数部分的估计量,针对系数函数是否为常数进行结合Bootstrap方法的广义似然检验,并在适当条件下证明所得结果的渐近性质。对带有不可加测量误差的模型,我们介绍了基于最优差分序列的纠正局部多项式估计和纠偏的一元化估计两种估计方法。在适当的假设条件下,分别证明两种方法所得参数和系数函数的估计量的渐近性质。最后,通过相关数值模拟与实例数据验证文中所提出的估计方法以及检验方法的有效性。
[Abstract]:Regression model is one of the most important models in statistics. As an important branch of regression model, semi-parametric model has attracted much attention because of its combination of parametric model and non-parametric model. The model combines parametric components and nonparametric functions to fully mine the information between the data so as to maximize the fitting effect of the model. As one of the semi-parametric models, the partial nonlinear variable coefficient model inherits the characteristics of the semi-parametric model, which not only reduces the deviation of the model, but also avoids the "dimension curse". These characteristics make the model more applicable to practical problems. However, in solving practical problems, there are often a variety of reasons for the data can not be accurately observed. For example, the tools or operations used in the measurement are not standardized, the impact of the external environment, sampling, etc. In statistical research, we usually refer to the problem with measurement error as the "measurement error problem", and call the model with measurement error as "measurement error model". In this paper, we mainly study the partial nonlinear variable coefficient model with measurement error, in which the nonparametric covariables have errors, and the errors can be divided into additive and non-additive cases. For the model with additive measurement error, we use the rectified contour least square to obtain the estimators of the parametric and non-parametric parts, and to test whether the coefficient function is constant or not, the generalized likelihood test combined with the Bootstrap method is carried out. The asymptotic properties of the obtained results are proved under appropriate conditions. For the model with non-additive measurement error, we introduce two estimation methods based on the optimal difference sequence: correcting local polynomial estimation and rectifying the error. Under appropriate assumptions, the asymptotic properties of the estimators of the parameters and coefficient functions obtained by the two methods are proved respectively. Finally, the validity of the proposed estimation method and the test method are verified by the correlation numerical simulation and the example data.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1

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