Efficient estimation of pathwise differentiable target parameters with the undersmoothed highly adaptive lasso

• Type of Material: Article
• Authors: Mark J van der Laan, University of California Berkeley; David Benkeser, Emory University; Weixin Cai, University of California Berkeley
• Language: English
• Date: 2022-07-15
• Publisher: WALTER DE GRUYTER GMBH
• Publication Version: Author Accepted Manuscript (After Peer Review)
• Copyright Statement: © 2022 Walter de Gruyter GmbH, Berlin/Boston
• License: Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International
• Final Published Version (URL): http://dx.doi.org/10.1515/ijb-2019-0092
• Title of Journal or Parent Work: INTERNATIONAL JOURNAL OF BIOSTATISTICS
• Volume: 19
• Issue: 1
• Start Page: 261
• End Page: 289
• Grant/Funding Information: This work was supported by the National Institute of Allergy and Infectious Diseases (grant number 5R01AI074345-09).
• Keywords: Science & Technology; Mathematics; undersmoothing; sectional variation norm; Life Sciences & Biomedicine; Mathematical & Computational Biology; asymptotically efficient estimator; canonical gradient; cross-validation; Physical Sciences; highly adaptive lasso; Statistics & Probability
• Research Categories: Biology, Bioinformatics; Biology, Biostatistics

Abstract

We consider estimation of a functional parameter of a realistically modeled data distribution based on observing independent and identically distributed observations. The highly adaptive lasso estimator of the functional parameter is defined as the minimizer of the empirical risk over a class of cadlag functions with finite sectional variation norm, where the functional parameter is parametrized in terms of such a class of functions. In this article we establish that this HAL estimator yields an asymptotically efficient estimator of any smooth feature of the functional parameter under a global undersmoothing condition. It is formally shown that the L 1-restriction in HAL does not obstruct it from solving the score equations along paths that do not enforce this condition. Therefore, from an asymptotic point of view, the only reason for undersmoothing is that the true target function might not be complex so that the HAL-fit leaves out key basis functions that are needed to span the desired efficient influence curve of the smooth target parameter. Nonetheless, in practice undersmoothing appears to be beneficial and a simple targeted method is proposed and practically verified to perform well. We demonstrate our general result HAL-estimator of a treatment-specific mean and of the integrated square density. We also present simulations for these two examples confirming the theory.