UNIFORMLY VALID POST-REGULARIZATION CONFIDENCE REGIONS FOR MANY FUNCTIONAL PARAMETERS IN Z-ESTIMATION FRAMEWORK

• 成果类型: Article
• 署名作者: Belloni, Alexandre; Chernozhukov, Victor; Chetverikov, Denis; Wei, Ying
• 署名单位: Duke University; Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT); University of California System; University of California Los Angeles; Columbia University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1671
• 发表日期: 2018
• 页码: 3643-3675
• 关键词: high-dimensional regression; SQUARE-ROOT LASSO; model-selection; linear-models; sparse models; semiparametric regression; econometric-models; EFFICIENCY BOUNDS; inference; approximation

摘要

In this paper, we develop procedures to construct simultaneous confidence bands for (p) over tilde 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) over tilde >> 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.