Targeted Inference Involving High-Dimensional Data Using Nuisance Penalized Regression

Analysis of high-dimensional data has received considerable and increasing attention in statistics. In practice, we may not be interested in every variable that is observed. Instead, often some of the variables are of particular interest, and the remaining variables are nuisance. To this end, we propose the nuisance penalized regression which does not penalize the parameters of interest. When the coherence between interest parameters and nuisance parameters is negligible, we show that resulting estimator can be directly used for inference without any correction. When the coherence is not negligible, we propose an iterative procedure to further refine the estimate of interest parameters, based on which we propose a modified profile likelihood based statistic for hypothesis testing. The utilities of our general results are demonstrated in three specific examples. Numerical studies lend further support to our method.高维数据分析在统计学领域已受到广泛且日益增长的关注。实际应用中，我们未必对所有观测变量均感兴趣，通常仅聚焦部分核心变量，剩余变量则属于扰动变量（nuisance variable）。为此，我们提出了扰动惩罚回归（nuisance penalized regression）方法，该方法不对目标参数（parameters of interest）施加惩罚。当目标参数与扰动参数（nuisance parameter）之间的相干性可忽略时，我们证明所得估计量可直接用于统计推断，无需任何校正。当相干性不可忽略时，我们提出一种迭代流程以进一步优化目标参数的估计，并基于该流程提出了用于假设检验的改进轮廓似然（profile likelihood）统计量。我们通过三个具体实例验证了所提通用结论的实用性，数值仿真实验进一步为该方法提供了有力支撑。