Structured estimation for the nonparametric Cox model

Rui Song,Jelena Bradic

doi:10.1214/15-ejs1004

Abstract

To better understand the interplay of censoring and sparsity we develop finite sample properties of nonparametric Cox proportional hazard's model. Due to high impact of sequencing data, carrying genetic information of each individual, we work with over-parametrized problem and propose general class of group penalties suitable for sparse structured variable selection and estimation. Novel non-asymptotic sandwich bounds for the partial likelihood are developed. We establish how they extend notion of local asymptotic normality (LAN) of Le Cam's. Such non-asymptotic LAN principles are further extended to high dimensional spaces where $p \gg n$. Finite sample prediction properties of penalized estimator in non-parametric Cox proportional hazards model, under suitable censoring conditions, agree with those of penalized estimator in linear models.

Highlights

Prediction of an instantaneous rate of occurrence of events when covariates are high dimensional plays a critical role in contemporary genetics studies underlying the causes of many incurable diseases
As with all asymptotic statistical properties, it is important to assess its relevance to the finite sample regime
Huang et al (2013) show non-asymptotic oracle estimation error bounds for the Cox model and with the LASSO penalty

Summary

Introduction

Prediction of an instantaneous rate of occurrence of events when covariates are high dimensional plays a critical role in contemporary genetics studies underlying the causes of many incurable diseases. Among the first theoretical work on the Cox model with right censored data that allows p ≫ n is Bradic et al (2011), where the authors documented good asymptotic variable selection properties of the LASSO and SCAD penalty. Huang et al (2013) show non-asymptotic oracle estimation error bounds for the Cox model and with the LASSO penalty. One of the first work that addresses non-asymptotic oracle bounds for the nonparametric Cox model is Letue (2000) where author exemplifies importance of asymptotic versus non-asymptotic theory. Kong and Nan (2012) analyze non-asymptotic oracle prediction bounds for the additive Cox model with fixed design, whereas Gaıffas and Guilloux (2012) analyze non-asymptotic oracle estimation bounds for the additive hazards models. Lemler (2012) shows non-asymptotic oracle bounds for the baseline hazards function in the additive Cox model with fixed design.

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