SPARSISTENCY AND RATES OF CONVERGENCE IN LARGE COVARIANCE MATRIX ESTIMATION

• 成果类型: Article
• 署名作者: Lam, Clifford; Fan, Jianqing
• 署名单位: University of London; London School Economics & Political Science; Princeton University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS720
• 发表日期: 2009
• 页码: 4254-4278
• 关键词: nonconcave penalized likelihood; longitudinal data; variable selection; oracle properties; Lasso; models; regression

摘要

This paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions. Here, sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one. Depending on the case of applications, sparsity priori may occur on the covariance matrix, its inverse or its Cholesky decomposition. We study these three sparsity exploration problems under a unified framework with a general penalty function. We show that the rates of convergence for these problems under the Frobenius norm are of order (s(n) log p(n)/n)(1/2), where s(n) is the number of nonzero elements, p(n) is the size of the covariance matrix and n is the sample size. This explicitly spells out the contribution of high-dimensionality is merely of a logarithmic factor. The conditions on the rate with which the tuning parameter lambda(n) goes to 0 have been made explicit and compared under different penalties. As a result, for the L(1)-penalty, to guarantee the sparsistency and optimal rate of convergence, the number of nonzero elements should be small: s'(n) = O(p(n)) at most, among O(p(n)(2)) parameters, for estimating sparse covariance or correlation matrix, sparse precision or inverse correlation matrix or sparse Cholesky factor, where s'(n) is the number of the nonzero elements on the off-diagonal entries. On the other hand, using the SCAD or hard-thresholding penalty functions, there is no such a restriction.