变系数部分线性模型的统计推断

发布时间：2018-02-25 22:32

本文关键词： 变系数部分线性模型有效估计模型识别变量选择超高维　出处：《南京信息工程大学》2015年硕士论文　论文类型：学位论文

【摘要】：随着互联网数据库的不断扩展,实际问题中收集到的数据很多都是高维数据。为了处理高维数据分析的问题,许多参数和半参数模型被提出用来避免“维数祸根”的问题。在众多的参数和半参数模型中,变系数部分线性模型由于模型本身既包含了常系数,又包含了函数型系数,从而受到广泛的关注。一般常用的估计模型方法有最小二乘法(LSE),最小方差法(MAVE)等,但这些方法得到的估计结果可能不是有效估计,这就要求我们发展更适合的估计方法。另一方面,由于数据的不断积累,数据中的协变量的数目经常呈多项式速率增长,有时甚至呈指数速率增长。在高维数据下如何更好地对参数或半参数模型进行估计和统计推断,就更加重要。因此,当我们研究变系数部分线性模型在高维和超高维情况下的统计问题时,就要求我们发展更合适的处理方法。本文系统地研究了变系数部分线性模型的估计、变量选择以及在超高维数据中的降维问题。研究结果表明通过建立有效估计方程可以得到有效估计；使用group lasso方法进行变量选择可以识别出常系数变量和函数型变量；通过排序KL距离来进行特征筛选,可以降低超高维数据的维数。本文研究了变系数部分线性模型在不同维数情况下的一些统计问题。主要内容如下：(1)研究了带异方差的变系数部分线性模型的估计有效性问题,给出完全样本下感兴趣参数的有效得分向量函数和有效估计。提出有效估计方程并给出带异方差的变系数部分线性模型的半参数有效界,证明所得估计为有效估计,并证明其大样本性质,通过数值模拟研究其有限样本性质。(2)研究了在高维数据下,变系数部分线性模型的变量选择问题。提出两阶段变量选择方法,分别对模型的线性部分和变系数部分做变量选择,得到参数的Adaptive Lasso估计,证明了估计的渐近性质与相合性,并利用数值模拟研究估计的有限样本性质。(3)研究了超高维数据情况下的变系数模型变量筛选问题,提出基于KL距离的变量筛选方法,通过条件累计分布函数构造协变量与响应变量之间的边际KL距离统计量,进行排序来筛选变量,并利用数值模拟验证了所提出方法的有限样本性质。
[Abstract]:With the continuous expansion of the Internet database, many of the data collected in practical problems are high-dimensional data. In order to deal with the problem of high-dimensional data analysis, Many parametric and semi-parametric models have been proposed to avoid the problem of "dimensionality curse". Among many parametric and semi-parametric models, the partial linear model with variable coefficients contains both constant and functional coefficients. The commonly used estimation methods are the least square method (LSE), the least variance method (MAVEV) and so on, but the estimation results obtained by these methods may not be valid. This requires us to develop more appropriate estimation methods. On the other hand, because of the constant accumulation of data, the number of covariables in the data often increases at a polynomial rate. Sometimes it even increases exponentially. How to better estimate and statistically infer parametric or semi-parametric models under high-dimensional data is even more important. When we study the statistical problems of partial linear models with variable coefficients in high and ultra-high dimensions, we are required to develop more suitable methods of dealing with them. In this paper, the estimation of partial linear models with variable coefficients is studied systematically. Variable selection and dimensionality reduction in ultra-high dimensional data. The results show that effective estimation can be obtained by establishing effective estimation equations, constant coefficient variables and functional variables can be identified by group lasso method. By sorting the KL distance for feature filtering, In this paper, we study some statistical problems of partial linear models with variable coefficients under different dimensions. The main contents are as follows: (1) the estimation validity of partial linear models with variable coefficients with heteroscedasticity is studied. The effective score vector function and effective estimate of the parameters of interest under complete samples are given. The effective estimation equation and the semi-parametric efficient bound of the partial linear model with variable coefficients with heteroscedasticity are proposed, and the obtained estimates are proved to be effective estimates. The problem of variable selection for variable coefficient partial linear model under high-dimensional data is studied by numerical simulation. A two-stage variable selection method is proposed. The linear part and variable coefficient part of the model are selected, and the Adaptive Lasso estimation of the parameters is obtained. The asymptotic property and consistency of the estimator are proved. The variable selection problem of variable coefficient model under ultra-high dimensional data is studied by using finite sample properties of numerical simulation. A variable selection method based on KL distance is proposed. The marginal KL distance statistics between the covariable and the response variable are constructed by the conditional cumulative distribution function and sorted to filter the variables. The finite sample properties of the proposed method are verified by numerical simulation.
【学位授予单位】：南京信息工程大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O212.1

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