基于剖面似然的统计推断

The profile likelihood is a commonly used method for statistical inference for different statistical models. It could yield efficient estimator and is computationally efficient. We study the profile likelihood method in two aspects for the present project. First, variable selection plays an important role in high dimensional statistical modeling which nowadays appears in many areas and is key to various scientific discoveries, Chen, Tang, Gao & Shi (2014) proposed the penalized profile likelihood method for linear regression models, which performs robustly well without error distribution specification. However, they assumed that the dimension of the covariates is fixed. We will study the statistical theory of the penalized profile likelihood when the dimension of the covariates is either high or ultrahigh in this project, on which practice relies heavily to justify its validity. Second, longitudinal data arise frequently in biomedical and health studies in which repeated measurements from the same subject are correlated. Based on the methodology of profile likelihood in Chen, Tang, Gao & Shi (2014), we propose a new method for longitudinal data analysis, which works without any distributional assumption on the error, hence, is adaptive and flexible. The newly developed method takes into account the within-subject correlation properly thus could yield efficient estimators. The asymptotic properties of the estimates will be investigated.

在各种统计模型中，剖面似然是一种常用的统计推断方法，它计算简便且能得到有效估计。本项目将从两大方面研究剖面似然。首先，变量选择在出现于诸多领域的高维统计模型中起着重要作用，且在各种科学研究中扮演关键角色，针对线性模型，Chen, Tang, Gao & Shi (2014)提出了惩罚剖面似然方法，它无需误差分布假设，表现稳健。然而他们假设协变量维数固定。统计渐进性质是高维实际数据分析的理论支撑，本项目将分别研究惩罚剖面似然在高维和超高维协变量情形下的统计理论。其次，纵向数据经常出现在生物医学和健康研究中，在这些研究中同一个体的多次观测是相关的。基于剖面似然方法(Chen, Tang, Gao & Shi, 2014)，我们给出一种纵向数据分析新方法。该方法无需误差分布假设，因而很灵活且实用性强；它合理地将个体内部相关性融入估计过程，使得所得估计是有效的。我们将研究所得估计的渐进性质。

在各种统计模型中，核回归方法和剖面似然是一种常用的统计推断方法。本项目将从四个具体的方面着手研究。首先，纵向数据经常出现在生物医学和健康研究中，在这些研究中同一个体的多次观测是相关的。基于剖面似然方法，我们给出了一种纵向数据分析新方法。该方法无需误差分布假设，因而很灵活且实用性强；它合理地将个体内部相关性融入估计过程，使得所得估计是有效的。其次，统计渐进性质是高维实际数据分析的理论支撑，本项目研究了惩罚剖面似然在高维协变量情形下的统计理论。第三，本项目研究了基于核回归方法的超高维协方差矩阵估计及其性质。第四，研究了基于核回归方法的正定协方差矩阵估计及其性质。