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Uniformly valid post-regularization confidence regions for many functional parameters in z-estimation framework

Author(s)
Belloni, Alexandre; Chernozhukov, Victor V; Chetverikov, Denis; Wei, Ying
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Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/
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Abstract
In this paper, we develop procedures to construct simultaneous confidence bands for p potentially infinite-dimensional parameters after model selection for general moment condition models where p is potentially much larger than the sample size of available data, n. This allows us to cover settings with functional response data where each of the p parameters is a function. The procedure is based on the construction of score functions that satisfy Neyman orthogonality condition approximately. The proposed simultaneous confidence bands rely on uniform central limit theorems for high-dimensional vectors (and not on Donsker arguments as we allow for p n). To construct the bands, we employ a multiplier bootstrap procedure which is computationally efficient as it only involves resampling the estimated score functions (and does not require resolving the high-dimensional optimization problems). We formally apply the general theory to inference on regression coefficient process in the distribution regression model with a logistic link, where two implementations are analyzed in detail. Simulations and an application to real data are provided to help illustrate the applicability of the results. Keyword: Inference after model selection ; moment condition models with a continuum of target parameters ; Lasso and Post-Lasso with functional response data
Date issued
2018-09
URI
https://hdl.handle.net/1721.1/122942
Department
Massachusetts Institute of Technology. Operations Research Center; Massachusetts Institute of Technology. Department of Economics
Journal
Annals of statistics
Publisher
Institute of Mathematical Statistics
Citation
Belloni, Alexandre et al. "Uniformly valid post-regularization confidence regions for many functional parameters in z-estimation framework." Annals of statistics 46, 6B (September 2018): 3643-3675 © 2019 Project Euclid
Version: Author's final manuscript
ISSN
0090-5364
Keywords
Statistics, Probability and Uncertainty, Statistics and Probability

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