Variance Estimation Using Refitted Cross-Validation in Ultrahigh Dimensional Regression

• 成果类型: Article
• 署名作者: Fan, Jianqing; Guo, Shaojun; Hao, Ning
• 署名单位: Princeton University; Chinese Academy of Sciences; University of Arizona
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/j.1467-9868.2011.01005.x
• 发表日期: 2012
• 页码: 37-65
• 关键词: nonconcave penalized likelihood; variable selection; model selection; Lasso; RECOVERY; sparsity

摘要

Variance estimation is a fundamental problem in statistical modelling. In ultrahigh dimensional linear regression where the dimensionality is much larger than the sample size, traditional variance estimation techniques are not applicable. Recent advances in variable selection in ultrahigh dimensional linear regression make this problem accessible. One of the major problems in ultrahigh dimensional regression is the high spurious correlation between the unobserved realized noise and some of the predictors. As a result, the realized noises are actually predicted when extra irrelevant variables are selected, leading to a serious underestimate of the level of noise. We propose a two-stage refitted procedure via a data splitting technique, called refitted cross-validation, to attenuate the influence of irrelevant variables with high spurious correlations. Our asymptotic results show that the resulting procedure performs as well as the oracle estimator, which knows in advance the mean regression function. The simulation studies lend further support to our theoretical claims. The naive two-stage estimator and the plug-in one-stage estimators using the lasso and smoothly clipped absolute deviation are also studied and compared. Their performances can be improved by the refitted cross-validation method proposed.