Boosted nonparametric hazards with time-dependent covariates

Donald K.K. Lee, Ningyuan Chen, Hemant Ishwaran

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)2101-2128
Number of pages28
JournalAnnals of Statistics
Issue number4
StatePublished - Aug 2021


  • Functional data
  • Gradient boosting
  • Likelihood functional
  • Regression trees
  • Stepsize shrinkage
  • Survival analysis

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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