BOOSTED NONPARAMETRIC HAZARDS WITH TIME-DEPENDENT COVARIATES

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 nonp...

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Bibliographic Details
Published inThe Annals of statistics Vol. 49; no. 4; p. 2101
Main Authors Lee, Donald K K, Chen, Ningyuan, Ishwaran, Hemant
Format Journal Article
LanguageEnglish
Published United States 01.08.2021
Subjects
gradient boosting
Primary 62N02
survival analysis
likelihood functional
functional data
regression trees
secondary 62G05,90B22
step-size shrinkage
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Summary: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 step-size 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 step-size restriction is a mechanism for preventing the curvature of the risk from derailing convergence.
ISSN:0090-5364
DOI:10.1214/20-AOS2028