广义函数型回归模型及缺失数据模型

发布时间：2018-01-25 15:08

本文关键词： 函数型数据分析函数型主成分分析广义函数型线性模型缺失数据随机缺失多项式样条渐近正态性　出处：《云南大学》2016年博士论文　论文类型：学位论文

【摘要】：随着数据收集工具和存储技术的进步,如何从海量的数据中挖掘出有价值的信息已成为各领域共同关注的问题。而作为一种能够处理高维复杂数据的统计方法-函数型数据分析方法,已成为近十多年来统计学领域研究的热点问题之一,并被广泛的应用到金融工程、生物化学工程、医学、水文、脑图像分析与建模等领域。缺失数据在实际中非常普遍,如在抽样调查中,被调查者不愿意回答某些问题,药物追踪试验中病人中途退出试验。在参数和半参数回归模型下,对缺失数据的处理与分析已被广泛研究,然而在函数型数据下,关于缺失数据问题的研究还非常少。另外,广义线性模型由于其能对各种不同的响应变量进行回归建模,因而自提出便被广泛研究。在函数型数据下,James (2002)首次将传统的广义线性模型推广到函数型数据,提出了广义函数型线性模型,并基于函数型主成分方法讨论了模型的统计推断问题,但用多项式样条方法来研究该模型的统计推断还比较少。为此,本文首先研究了广义函数型回归模型的多项式样条估计问题,其次讨论了响应变量随机缺失的函数型部分线性模型的统计推断问题,这些研究具有重要的理论意义和一定的实际应用价值。本文的主要研究工作包括：(1)在广义函数型线性模型下通过引入额外的协变量,提出了广义部分函数型线性模型,该模型是Zhang et al. (2007)和Shin (2009)的部分函数型线性模型的推广；构造了模型参数向量和斜率函数的多项式样条估计,并在一定条件下得到了参数估计的渐近正态性和斜率函数多项式样条估计的全局收敛速度。通过数值模拟说明了多项式样条方法的可行性。(2)在响应变量随机缺失的情况下研究了Lian(2011)提出的函数型部分线性模型的多项式样条估计,并对均值提出了基于回归插补和部分逆概率加权两种估计方法。在给定的条件下,得到了基于不完全数据模型斜率函数估计的收敛速度,以及两种均值估计的收敛速度。通过与Ferraty et al (2013)的结论相比较,发现在函数型部分线性模型下,均值的两种插补估计达不到(?)的相合速度,除非模型中的斜率函数限制在一个有限维的函数空间。通过数值模拟,研究了多项式样条估计的有限样本行为。(3)在响应变量随机缺失的情况下构建了函数型部分线性模型的函数型主成分估计。同样地,对均值提出了基于回归插补和部分逆概率加权两种估计方法。在给定的条件下,得到了基于不完全数据模型斜率函数估计的收敛速度,该结论与Lian(2011)的结论完全一样。对于两种均值估计,得到了类似的收敛速度,该结果表明：在本文给定的条件下达不到Ferraty et al (2013)中的(?)相合速度。最后,通过两个数值模拟说明了函数型主成分方法和多项式样条方法的可行性。
[Abstract]:Along with the data collection tool and the storage technology advancement. How to mine valuable information from the massive data has become a common concern in all fields. As a statistical method which can deal with high-dimensional and complex data, the functional data analysis method. It has become one of the hot issues in the field of statistics in recent ten years, and has been widely used in financial engineering, biochemistry engineering, medicine, hydrology. In the field of brain image analysis and modeling, missing data is very common in practice. For example, in the sample survey, the respondents are unwilling to answer some questions. Under parametric and semi-parametric regression models, the treatment and analysis of missing data have been extensively studied, however, in functional data. There are few researches on missing data problem. In addition, generalized linear model is widely studied under functional data because of its ability to model different response variables. James / 2002) extended the traditional generalized linear model to the functional data for the first time, proposed the generalized functional linear model, and discussed the statistical inference problem of the model based on the functional principal component method. However, the statistical inference of the model is less studied by polynomial spline method. In this paper, we first study the polynomial spline estimation of generalized functional regression model. Secondly, the problem of statistical inference for function-type partial linear model with random absence of response variables is discussed. These studies have important theoretical significance and practical application value. The main research work in this paper includes the introduction of additional covariables in the generalized functional linear model. In this paper, a generalized partial functional linear model is proposed, which is a generalization of the partial functional linear model of Zhang et al. 2007) and Shin et al. 2009). The polynomial spline estimation of the parameter vector and slope function of the model is constructed. Under certain conditions, the asymptotic normality of parameter estimation and the global convergence rate of slope function polynomial spline estimator are obtained. The feasibility of polynomial spline method is illustrated by numerical simulation. In the case of random absence of response variables, the polynomial spline estimation of functional partial linear model proposed by Lianqian 2011) is studied. Two estimation methods based on regression interpolation and partial inverse probability weighting are proposed for the mean value. The convergence rate of slope function estimation based on incomplete data model is obtained under given conditions. And the convergence rate of the two mean estimators. Compared with the conclusion of Ferraty et al. 2013), it is found that in the partial linear model of function type. The two interpolation estimates of the mean value can not reach? Unless the slope function in the model is confined to a finite dimensional functional space. In this paper, we study the behavior of finite sample of polynomial spline estimator. We construct the function-type principal component estimation of functional partial linear model under the condition of random absence of response variables. Two estimation methods based on regression interpolation and partial inverse probability weighting are proposed for the mean value. The convergence rate of slope function estimation based on incomplete data model is obtained under given conditions. This conclusion is exactly the same as that of Lianqian 2011). For the two mean estimators, a similar convergence rate is obtained. The results show that the Ferraty et al. 2013 can not be obtained under the given conditions in this paper. Finally, the feasibility of the functional principal component method and the polynomial spline method are demonstrated by two numerical simulations.
【学位授予单位】：云南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O212.1

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