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December 2018 Uniformly valid post-regularization confidence regions for many functional parameters in z-estimation framework
Alexandre Belloni, Victor Chernozhukov, Denis Chetverikov, Ying Wei
Ann. Statist. 46(6B): 3643-3675 (December 2018). DOI: 10.1214/17-AOS1671


In this paper, we develop procedures to construct simultaneous confidence bands for ${\tilde{p}}$ potentially infinite-dimensional parameters after model selection for general moment condition models where ${\tilde{p}}$ 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 ${\tilde{p}}$ 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 ${{\tilde{p}}\gg n}$). 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.


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Alexandre Belloni. Victor Chernozhukov. Denis Chetverikov. Ying Wei. "Uniformly valid post-regularization confidence regions for many functional parameters in z-estimation framework." Ann. Statist. 46 (6B) 3643 - 3675, December 2018.


Received: 1 February 2016; Revised: 1 October 2017; Published: December 2018
First available in Project Euclid: 11 September 2018

zbMATH: 1407.62268
MathSciNet: MR3852664
Digital Object Identifier: 10.1214/17-AOS1671

Primary: 62-07
Secondary: 62H99

Rights: Copyright © 2018 Institute of Mathematical Statistics


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Vol.46 • No. 6B • December 2018
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