Abstract:
|
We develop a novel method to construct asymptotically honest confidence regions for high-dimensional linear models. The key idea is to divide regression parameters into strong and weak signals, and then construct confidence regions for each of them separately. This is achieved by a two-step approach, consisting of a projection step and a shrinkage step, of which the latter is implemented via Stein estimation. We establish that the diameter of the confidence region converges at the optimal rate for sparse coefficient vectors. We further generalize the two-step method to construct multiple confidence regions from the same data, allowing for choosing one of them via optimizing certain criterion while guaranteeing the desired asymptotic honesty. Through simulation studies, we demonstrate that our two-step Stein method outperforms other competitors, including an oracle method with prior knowledge about the true sparsity of the model. Finally, we propose a two-step lasso method that may achieve a faster convergence rate adaptive to a sparsity level given in priori.
|