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

Abstract

In this paper, we develop procedures to construct simultaneous confidence bands for potentially infinite-dimensional parameters after model selection for general moment condition models where 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 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 ). 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.