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August 2021 Boosted nonparametric hazards with time-dependent covariates
Donald K. K. Lee, Ningyuan Chen, Hemant Ishwaran
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Ann. Statist. 49(4): 2101-2128 (August 2021). DOI: 10.1214/20-AOS2028


Given functional data from a survival process with time-dependent covariates, we derive a smooth convex representation for its nonparametric log-likelihood functional and obtain its functional gradient. From this, we devise a generic gradient boosting procedure for estimating the hazard function nonparametrically. An illustrative implementation of the procedure using regression trees is described to show how to recover the unknown hazard. The generic estimator is consistent if the model is correctly specified; alternatively, an oracle inequality can be demonstrated for tree-based models. To avoid overfitting, boosting employs several regularization devices. One of them is stepsize restriction, but the rationale for this is somewhat mysterious from the viewpoint of consistency. Our work brings some clarity to this issue by revealing that stepsize restriction is a mechanism for preventing the curvature of the risk from derailing convergence.

Funding Statement

NC was supported by HKUST grant R9382 and NSERC Discovery Grant RGPIN-2020-04038. HI was supported by National Institutes of Health grants R01 GM125072 and R01 HL141892. DKKL was supported by a hyperplane.


The review team provided many insightful comments that significantly improved the paper. We are grateful to Brian Clarke, Jack Hall, Sahand Negahban and Hongyu Zhao for helpful discussions. Special thanks to Trevor Hastie for early formative discussions. The dataset used in Section 5 was kindly provided by Dr. Kito Lord.


Download Citation

Donald K. K. Lee. Ningyuan Chen. Hemant Ishwaran. "Boosted nonparametric hazards with time-dependent covariates." Ann. Statist. 49 (4) 2101 - 2128, August 2021.


Received: 1 June 2020; Published: August 2021
First available in Project Euclid: 29 September 2021

Digital Object Identifier: 10.1214/20-AOS2028

Primary: 62N02
Secondary: 62G05 , 90B22

Keywords: functional data , gradient boosting , likelihood functional , regression trees , stepsize shrinkage , Survival analysis

Rights: Copyright © 2021 Institute of Mathematical Statistics


Vol.49 • No. 4 • August 2021
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