Boosted Nonparametric Hazards with Time-Dependent Covariates
33 Pages Posted: 28 Jan 2017 Last revised: 29 Jun 2020
Date Written: February 12, 2017
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. We show that 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.
Keywords: survival analysis; gradient boosting; functional data; step-size shrinkage; regression trees; likelihood functional; queuing transition rates; emergency departments
JEL Classification: C14, C24, C34, C41, C44, C53
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