Boosting hazard regression with time-varying covariates
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Consider a left-truncated right-censored survival process whose evolution depends on time-varying covariates. Given functional data samples from the process, we propose a gradient boosting procedure for estimating its log-intensity function in a flexible manner to capture time-covariate interactions. The estimator is shown to be consistent if the model is correctly specified. Alternatively an oracle inequality can be demonstrated for tree-based models. We use the procedure to shed new light on a question from the operations literature concerning the effect of workload on service rates in an emergency department. 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: In theoretical treatments of classification and regression problems, unrestricted greedy step-sizes appear to suffice. Given that the partial log-likelihood functional for hazard regression has unbounded curvature, our study suggests that step-size restriction might be a mechanism for preventing the curvature of the risk from derailing convergence.
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