Semi-Low-Dimensional Inference Via Bias Correction
We consider statistical inference in a semi-low-dimensional approach to the analysis of high-dimensional data. The relationship between this semi-low-dimensional approach and regularized estimation of high-dimensional objects is parallel to the more familiar one between semi-parametric analysis and nonparametric estimation. Low-dimensional projection methods are used to correct the bias of regularized high-dimensional estimators, leading to efficient point and interval estimation. In linear regression and Gaussian graphical models, the Stein method can be used to remove the bias of regularized estimators, leading to sharp sample size requirement for de-biasing the Lasso and sharp uniform error bounds for the Lasso estimator itself.