含非线性函数的半参数回归模型的经验似然推断

发布时间：2018-04-17 08:02

本文选题：半参数回归模型 + 非线性函数　；参考：《西北工业大学》2015年博士论文

【摘要】：含非线性函数的半参数回归模型具有参数模型的可解释性和非参数模型的灵活性,并且克服了线性函数在表述客观模型方面的局限性.在应用中经常遇到缺失数据和测量误差数据等复杂数据.经验似然方法作为一种重要的非参数统计方法,广泛应用于兴趣参数的置信域构造.因此,在复杂数据下研究含非线性函数的半参数回归模型的经验似然推断具有一定的理论意义和实用价值.本论文研究了部分非线性模型和变系数部分非线性模型的经验似然推断,还研究了时空变系数模型的非平稳性检验问题.本文的主要研究成果如下:(1)首次给出了部分非线性模型的经验似然推断.构造了模型中未知参数和未知函数的对数经验似然比函数,证明了其渐近χ2分布,给出了非线性函数中未知参数的置信域和未知系数函数的置信带.同时,得到了未知参数和未知函数的极大经验似然估计,证明了估计量的渐近正态性.模拟结果和实例分析表明在构造未知参数的置信区间和未知函数的置信带方面,经验似然方法优于近似正态方法.(2)探索了复杂数据下部分非线性模型中未知参数的经验似然置信域估计问题.对于响应变量随机缺失的部分非线性模型,为了避免已有文献中权重因子和调整因子的估计,提出了一种局部纠偏的线性插补技术,提高了估计的精度.对于变量带测量误差的部分非线性模型,借助替代数据和核实样本,给出了未知参数的两种估计.构造了未知参数的对数似然比函数,证明了其渐近于加权的χ2分布之和.模拟结果表明在构造未知参数的置信区间方面,经验似然方法优于近似正态方法.(3)首次给出了变系数部分非线性模型的经验似然推断.构造了未知参数的对数经验似然比函数,证明了其渐近χ2分布,得到了未知参数的极大经验似然估计,并证明了其渐近正态性.对于非参数部分带测量误差的变系数部分非线性模型,提出了未知参数的局部纠偏的剖面最小二乘估计,证明了其渐近正态性,构造了未知参数的对数经验似然比函数,并证明了其渐近χ2分布.模拟结果验证了方法的有效性.(4)检验了时空变系数回归模型的非平稳性.构造了广义似然比统计量,对整个回归关系进行关于时间和空间的非平稳性检验,利用Bootstrap方法计算检验的p值;构造了合适的统计量对各个回归系数进行关于时间和空间的非平稳性检验,利用三阶矩χ2逼近方法计算检验的p值.模拟算例和实际例子表明检验方法的有效性.
[Abstract]:The semi-parametric regression model with nonlinear function has the interpretability of parametric model and the flexibility of non-parametric model, and overcomes the limitation of linear function in describing objective model.Complex data such as missing data and measuring error data are often encountered in applications.As an important nonparametric statistical method, empirical likelihood method is widely used in constructing confidence regions of parameters of interest.Therefore, it is of theoretical significance and practical value to study the empirical likelihood inference of semi-parametric regression models with nonlinear functions under complex data.In this paper, empirical likelihood inference for partial nonlinear models and partial nonlinear models with variable coefficients is studied, and the nonstationarity test of spatio-temporal models with variable coefficients is also studied.The main results of this paper are as follows: 1) for the first time, the empirical likelihood inference of partial nonlinear model is given.The logarithmic empirical likelihood ratio function of unknown parameter and unknown function in the model is constructed, and its asymptotic 蠂 2 distribution is proved. The confidence region of unknown parameter and the confidence band of unknown coefficient function in nonlinear function are given.At the same time, the maximum empirical likelihood estimators of unknown parameters and unknown functions are obtained, and the asymptotic normality of the estimators is proved.The results of simulation and the analysis of examples show that in the construction of confidence intervals of unknown parameters and confidence bands of unknown functions,The empirical likelihood method is superior to the approximate normal method. 2) the empirical likelihood confidence region estimation problem of unknown parameters in partial nonlinear models with complex data is explored.In order to avoid the estimation of weight factor and adjustment factor in some nonlinear models with random missing response variables, a local correction linear interpolation technique is proposed to improve the accuracy of the estimation.For the partial nonlinear model with variable measurement error, two kinds of estimations of unknown parameters are given by means of substitute data and verified samples.The logarithmic likelihood ratio function of unknown parameters is constructed, and it is proved that it is asymptotically equal to the sum of weighted 蠂 ~ 2 distributions.The simulation results show that the empirical likelihood method is superior to the approximate normal method in constructing confidence intervals of unknown parameters.The logarithmic empirical likelihood ratio function of unknown parameters is constructed, its asymptotic 蠂 ~ 2 distribution is proved, the maximum empirical likelihood estimation of unknown parameters is obtained, and its asymptotic normality is proved.For the nonparametric partial nonlinear model with variable coefficients with measurement errors, the least square estimation of local correction of unknown parameters is presented, its asymptotic normality is proved, and the logarithmic empirical likelihood ratio function of unknown parameters is constructed.The asymptotic 蠂 2 distribution is proved.The simulation results verify the validity of the method and test the nonstationarity of the spatiotemporal variable coefficient regression model.The generalized likelihood ratio statistic is constructed and the non-stationary test of time and space is carried out for the whole regression relationship. The p value of the test is calculated by using Bootstrap method.The nonstationary test of time and space for each regression coefficient is constructed, and the p value of the test is calculated by using the third-order moment 蠂 ~ 2 approximation method.Simulation examples and practical examples show the effectiveness of the test method.
【学位授予单位】：西北工业大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O212.1

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