Estimation Theory for Semiparametric Models with Bundled Parameters

在参数分离的半参数模型中推广了已有的关于M-估计和Z-估计的渐近分布理论，以适应半参数模型的估计准则被参数化为捆绑参数的情形，即无限维参数是感兴趣参数的未知函数。这一建议是基于几种不同截尾机制下截尾数据线性回归模型的有效估计、单指数模型的有效估计、带未知连接函数的Cox回归模型的部分似然估计以及生存分析中缺失数据的加权估计等几个统计问题提出的。研究人员还建议将参数捆绑的一般理论应用于所有这些问题，特别是在无限维扰扰参数被回归样条逼近的情况下。所提出的研究的主要动机是PI在生物医学研究中的合作，在生物医学研究中，需要更健壮的统计建模技术来减少模型错误指定的不确定性，特别是当数据由于研究跟踪有限而不完整的时候。这项研究还将使研究人员能够在高级生存分析课程中加入更全面的统计结果，并有助于将半参数模型专题课程发展为常规的博士水平课程。拟议的研究活动将激励研究生成为能够从事基础统计研究的独立研究人员。

The investigator proposes extensions of existing asymptotic distributional theories for M- and Z-estimations in the semiparametric models with separated parameters to accommodate situations where the estimation criteria for the semiparametric models are parameterized with bundled parameters, i.e. the infinite dimensional parameter is an unknown function of the parameter of interest. The proposal is motivated by several statistical problems including the efficient estimation in the linear regression model with censored data under several different censoring mechanisms, the efficient estimation in the single index model, the partial likelihood estimation in the Cox regression model with an unknown link function, and the weighted estimation for missing data problems in survival analysis. The investigator also proposes to apply the general theory for bundled parameters to all these problems, particularly for the case that the infinite dimensional nuisance parameter is approximated by regression splines.The proposed research is primarily motivated by PI's collaboration in biomedical studies, where more robust statistical modeling techniques are desirable to reduce the uncertainty of model misspecification, particularly when data are incomplete due to limited study follow-up. The proposed research will also allow the investigator to add more thorough statistical results to the course of advanced survival analysis and be helpful in developing the special topic course on semiparametric models into a regular Ph.D. level course. The proposed research activities will motivate graduate students to become independent researchers who are able to engage in fundamental statistical research.