ADAPTIVE ESTIMATION OF THE RANK OF THE COEFFICIENT MATRIX IN HIGH-DIMENSIONAL MULTIVARIATE RESPONSE REGRESSION MODELS

• 成果类型: Article
• 署名作者: Bing, Xin; Wegkamp, Marten H.
• 署名单位: Cornell University; Cornell University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1774
• 发表日期: 2019
• 页码: 3157-3184
• 关键词: selection; EQUATIONS

摘要

We consider the multivariate response regression problem with a regression coefficient matrix of low, unknown rank. In this setting, we analyze a new criterion for selecting the optimal reduced rank. This criterion differs notably from the one proposed in Bunea, She and Wegkamp (Ann. Statist. 39 (2011) 1282-1309) in that it does not require estimation of the unknown variance of the noise, nor does it depend on a delicate choice of a tuning parameter. We develop an iterative, fully data-driven procedure, that adapts to the optimal signal-to-noise ratio. This procedure finds the true rank in a few steps with overwhelming probability. At each step, our estimate increases, while at the same time it does not exceed the true rank. Our finite sample results hold for any sample size and any dimension, even when the number of responses and of covariates grow much faster than the number of observations. We perform an extensive simulation study that confirms our theoretical findings. The new method performs better and is more stable than the procedure of Bunea, She and Wegkamp (Ann. Statist. 39 (2011) 1282-1309) in both low- and high-dimensional settings.