Covariate assisted screening and estimation

Abstract

Consider a linear model $Y=X\beta+z$, where $X=X_{n,p}$ and $z\sim N(0,I_{n})$. The vector $\beta$ is unknown but is sparse in the sense that most of its coordinates are $0$. The main interest is to separate its nonzero coordinates from the zero ones (i.e., variable selection). Motivated by examples in long-memory time series (Fan and Yao [Nonlinear Time Series: Nonparametric and Parametric Methods (2003) Springer]) and the change-point problem (Bhattacharya [In Change-Point Problems (South Hadley, MA, 1992) (1994) 28–56 IMS]), we are primarily interested in the case where the Gram matrix $G=X'X$ is nonsparse but sparsifiable by a finite order linear filter. We focus on the regime where signals are both rare and weak so that successful variable selection is very challenging but is still possible.

We approach this problem by a new procedure called the covariate assisted screening and estimation (CASE). CASE first uses a linear filtering to reduce the original setting to a new regression model where the corresponding Gram (covariance) matrix is sparse. The new covariance matrix induces a sparse graph, which guides us to conduct multivariate screening without visiting all the submodels. By interacting with the signal sparsity, the graph enables us to decompose the original problem into many separated small-size subproblems (if only we know where they are!). Linear filtering also induces a so-called problem of information leakage, which can be overcome by the newly introduced patching technique. Together, these give rise to CASE, which is a two-stage screen and clean [Fan and Song Ann. Statist. 38 (2010) 3567–3604; Wasserman and Roeder Ann. Statist. 37 (2009) 2178–2201] procedure, where we first identify candidates of these submodels by patching and screening, and then re-examine each candidate to remove false positives.

For any procedure $\hat{\beta}$ for variable selection, we measure the performance by the minimax Hamming distance between the sign vectors of $\hat{\beta}$ and $\beta$. We show that in a broad class of situations where the Gram matrix is nonsparse but sparsifiable, CASE achieves the optimal rate of convergence. The results are successfully applied to long-memory time series and the change-point model.