title: COVARIANCE ASSISTED SCREENING AND ESTIMATION.
abstract: Consider a linear modelY=Xβ +z, whereX=Xn,pandz~N(0,In). The vector β is unknown and it is of interest to separate its nonzero coordinates from the zero ones (i.e., variable selection). Motivated by examples in long-memory time series (Fan and Yao, 2003) and the change-point problem (Bhattacharya, 1994), we are primarily interested in the case where the Gram matrixG=X'Xisnon-sparsebutsparsifiableby a finite order linear filter. We focus on the regime where signals are bothrare and weakso that successful variable selection is very challenging but is still possible. We approach this problem by a new procedure called theCovariance 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 ofinformation leakage, which can be overcome by the newly introducedpatchingtechnique. Together, these give rise to CASE, which is a two-stage Screen and Clean (Fan and Song, 2010; Wasserman and Roeder, 2009) procedure, where we first identify candidates of these submodels bypatching and screening, and then re-examine each candidate to remove false positives. For any procedure β̂ for variable selection, we measure the performance by the minimax Hamming distance between the sign vectors of β̂ and β. We show that in a broad class of situations where the Gram matrix is non-sparse but sparsifiable, CASE achieves the optimal rate of convergence. The results are successfully applied to long-memory time series and the change-point model.
authors.affiliation: Princeton University and Carnegie Mellon University.
authors.forename: By Tracy
authors.initials: BT
authors.lastname: Ke
authors.affiliation: Princeton University and Carnegie Mellon University.
authors.forename: Jiashun
authors.initials: J
authors.lastname: Jin
authors.affiliation: Princeton University and Carnegie Mellon University.
authors.forename: Jianqing
authors.initials: J
authors.lastname: Fan
keywords: Asymptotic minimaxity; Graph of Least Favorables (GOLF); Graph of Strong Dependence (GOSD); Hamming distance; Rare and Weak signal model; multivariate screening; phase diagram; sparsity; variable selection