Higher Order Asymptotic Properties of the Bootstrap in Post Model Selection Inference in High Dimensions

• Activity Number: 599 - Resampling Methods for High-Dimensional Inference
• Type: Topic Contributed
• Date/Time: Thursday, August 1, 2019 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Soumendra N Lahiri*
• Companies: North Carolina State University
• Keywords: Bootstrap; penalized regression; Second order correctness

Abstract

Chatterjee and Lahiri (2013) showed that under suitable conditions, the residual Bootstrap is second order correct for studentized pivots based on the ALASSO. One of the major limitations of their result is the existence of a preliminary estimator satisfying certain probabilistic bounds that are hard to verify in the p>n case. In this talk, we show that the second order correctness property holds quite generally (including the p>n case) for a number of penalized regression methods satisfying a version of the Oracle property of Fan and Li (2001). In particular, we show that under some suitable conditions, some popular nonconvex penalization functions including the SCAD and the MCP also enjoy second order correctness. Further, the Bootstrap offers remarkably accurate approximation in the case of LASSO even in situations where the (normalized) LASSO estimator itself fails to converge to a proper limit law, so that the standard approach of limit distribution based calibration of tests and confidence intervals is not applicable.