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.