Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems

Journal Article (Journal Article)

We develop uniformly valid confidence regions for regression coefficients in a highdimensional sparse median regression model with homoscedastic errors. Our methods are based on amoment equation that is immunized against nonregular estimation of the nuisance part of the median regression function by using Neyman's orthogonalization. We establish that the resulting instrumental median regression estimator of a target regression coefficient is asymptotically normally distributed uniformly with respect to the underlying sparse model and is semiparametrically efficient.We also generalize our method to a general nonsmooth Z-estimation framework where the number of target parameters is possibly much larger than the sample size. We extend Huber's results on asymptotic normality to this setting, demonstrating uniform asymptotic normality of the proposed estimators over rectangles, constructing simultaneous confidence bands on all of the target parameters, and establishing asymptotic validity of the bands uniformly over underlying approximately sparse models.

Full Text

Duke Authors

Cited Authors

  • Belloni, A; Chernozhukov, V; Kato, K

Published Date

  • March 1, 2015

Published In

Volume / Issue

  • 102 / 1

Start / End Page

  • 77 - 94

Electronic International Standard Serial Number (EISSN)

  • 1464-3510

International Standard Serial Number (ISSN)

  • 0006-3444

Digital Object Identifier (DOI)

  • 10.1093/biomet/asu056

Citation Source

  • Scopus