高维复杂数据的统计推断及其应用研究

应用核估计、局部多项式估计、参数与非参数方法、经验似然、二次惩罚法与惩罚经验似然技术等以及综合应用其中的方法，包括创新一些更有效的推断方法，在观察数据（包括纵向数据）不完全下，讨论协变量具有高维（超高维）时分位数回归模型的估计、变量选择、经验似然及假设检验等，涉及的分位数回归模型包括非参数模型、部分线性变系模型、部分线性单指标模型、以及单指标系数模型等；对函数型数据，在缺失观察或（及）部分协变量具有高维（超高维）设置下，研究均值回归与分位数回归模型的估计与变量选择；对随机缺失数据或纵向数据，在协变量具有高维（超高维）情况下，讨论分位数回归模型中未知参数的贝叶斯分析，构造均值回归及分位数回归模型中参数的变点与序列变点的估计量并证明它们的的大样本性质。利用实际数据和模拟数据对理论结果做模拟验证，并探索理论结果在实际问题中的应用。

By using kernel estimation, local polynomial estimation, parameter and nonparametric methods, empirical likelihood, two-step penalized methods, penalized empirical likelihood technology as well as synthetical applications for these methods, inclouding some more effective and innovative inference methods, under observed data (including longitudinal data) are incomplete, we study estimation, variable selection, empirical likelihood and hypothesis test for quantile regression model with high-dimensional (ultra-high dimensional) covariates, where the quantile regression model includes nonparametric model, partially linear varying-coefficient model, partially linear single-index model, single-index coefficient model, etc. For functional data, the estimation and variable selection of mean regression and quantile regression models are discussed under missing observations or/and high-dimensional (super-high dimensional) for partial covariates. For random missing data or longitudinal data, when the covarites are high dimension (ultra-high dimension), we investigate Bayesian analysis of unknown parameters in the quantile regression models, and construct estimators of change point and sequential change point of parameters in the mean regression and quantile regression models and prove their large-sample properties. The obtained theoretical results are verified by real data analysis and simulation study, and applications of the theoretical results in practical problems are explored.