Adaptive estimation of the baseline hazard function in the Cox model by model selection, with high-dimensional covariates

Abstract

The purpose of this article is to provide an adaptive estimator of the baseline function in the Cox model with high-dimensional covariates. We consider a two-step procedure : first, we estimate the regression parameter of the Cox model via a Lasso procedure based on the partial log-likelihood, secondly, we plug this Lasso estimator into a least-squares type criterion and then perform a model selection procedure to obtain an adaptive penalized contrast estimator of the baseline function.Using non-asymptotic estimation results stated for the Lasso estimator of the regression parameter, we establish a non-asymptotic oracle inequality for this penalized contrast estimator of the baseline function, which highlights the discrepancy of the rate of convergence when the dimension of the covariates increases.

Machine Learning Methods for Septic Shock Prediction

“…to predict mortality in ICUs (Wang et al, 2015). Guilloux, Lemler, and Taupin (2016) used high-dimensional covariates with an adaptive estimator of the baseline function in the Cox model, which performed well with simulation data. Wu, Zheng, and Yu (2016) proposed a statistical method based on a semiparametric Logistic-Cox mixture model that worked reasonably for practical sample sizes.…”

“…The CPH model is used broadly in clinical studies for risk ratio estimation (Lin et al, 2013). This method had helped researchers achieve good results in medical predictions and risk estimations (Guilloux et al, 2016;Jackson & Cox, 2014;Tolosie & Sharma, 2014;Wang et al, 2014;Wang et al, 2015;Wu et al, 2016).…”

Non-parametric Poisson regression from independent and weakly dependent observations by model selection

“…Another point of origin for future research might be to consider the case of highdimensional covariates as was done recently in [GLT16a] and [GLT16b].…”

Adaptive kernel estimation of the baseline function in the Cox model with high-dimensional covariates

“…The selected bandwidthĥβ performs as well as the unknown oracle, up to the multiplicative constant C and up to a remaining term of order ln a (n) ln(pn k )/n, which is negligible. In Inequality (16), the infimum term is a classical one in such oracle inequalities for kernel estimators: the bias term ∥α h − α 0 ∥ 2 2 decreases when h decreases and the variance term V (h)…”

“…We establish the first adaptive and non-asymptotic oracle inequality, which guarantees the theoretical performance of this kernel estimator. The oracle inequality depends on non-asymptotic control of |β − β 0 | 1 deduced from an estimation inequality in Huang et al [17] and extended to the case of unbounded counting processes (see Guilloux et al [16] for details).…”