含指标项的变换模型的估计与经验似然分析

Linear transformation models are a class of linear regression models with an unknown increasing transformation for respose variable, in which the transformation preserves rankings for life times. Therefore the class of models can enjoy good conciseness and flexibility in survival data modeling. Based on censored survival data, we will generalize linear transformation models and study the semiparametric transformation models with index term. In this project, we mainly consider estimation and empirical likelihood inference for this class of models. This class of models have advantages of both linear transformation models and nonparametric transformation models. They not only can avoid "Curse of Dimensionality" but also can capture important features in high-dimensional data. By use of spline polynomial or locally polynomial approximation techniques and estimation function or maximum likelihood estimation methods, we study the estimation of unknown parameters and functions in these models. Moreover, we also give large sample properties and convergence ratio for all the estimates. By virtue of empirical likelihood approach, we will establish confidence interval for unknown parameters and confidence band for unknown functions in these models. At the same time, we will investigate the large sample properties of empirical likelihood ratio statistic. At last, we will apllied the theoretical results to the transformation models with index term to clinical medicine, economics studies, finance, biology etc. Through systemically studying this class of models, we not only enrich the theory of transformation models, but also broaden the application field of single index technique approach.

线性变换模型是一类响应变量为寿命时间未知单调递增变换的线性回归模型，该变换保持寿命时间的秩，从而该模型在寿命数据建模方面具有非常好的灵活性和简洁性。基于删失寿命数据，本项目拟推广线性变换模型，研究含有指标项的半参数变换模型的估计与经验似然分析问题。该类模型集线性变换模型和非参数变换模型优点于一身，既能避免"维数祸根"问题又能体现高维数据特征。利用样条多项式、局部多项式等逼近技术和估计方程、极大似然等估计方法，研究这类变换模型中未知参数和未知函数的估计问题及其大样本性质；通过经验似然方法，研究模型中未知参数的置信域和未知函数的置信带构造问题，同时给出经验似然比统计量的大样本性质；最后我们把该类模型的理论研究成果应用到临床医学，经济学，金融学，生物学等领域。通过系统研究此类模型不仅能够丰富变换模型的理论内容，而且能够拓广单指标技术研究方法的应用领域。

根据申请时的项目研究计划，我们对含指标项的半参数回归模型的估计与经验似然分析进行了研究。含指标项的回归模型是当前半参数统计模型应用研究的热点与重点问题之一，不仅能够降低线性模型的模型识别错误的风险而且能够避免非参数回归的“维数祸根”问题，而半参数变换模型是一类应用更广泛的生存分析模型，因此关于含指标项的半参数变换模型的研究具有重要的理论与现实意义。经过三年的努力，我们取得了如下成果：（1）借助未知指标函数B样条逼近和极大似然估计方法研究含单指标项的半参数变换模型的估计问题及相应大样本性质。（2）通过未知函数B样条逼近技术和部分似然，研究了含指标项的比例风险模型的经验似然置信域的构造问题以及相应的大样本性质；基于B样条、局部多项式逼近技术以及经验似然方法，研究了含指标项的模型误差序列相关性检验问题及其大样本性质。（3）通过B样条逼近，研究了广义可加变换模型的极大边际似然估计及相应的估计方法问题；基于B样条逼近的SCAD惩罚部分似然方法，研究了高维比例风险可加回归模型的模型选择问题。（4）把上述含指标项的半参数模型及相关统计分析方法推广到了医学、基因序列分析等应用领域问题的研究。