Oracle inequalities for weighted group lasso in high-dimensional misspecified Cox models

Yijun Xiao,Huiming Zhang,Ting Yan,Yuanyuan Zhang

doi:10.1186/s13660-020-02517-3

Abstract

We study the nonasymptotic properties of a general norm penalized estimator, which include Lasso, weighted Lasso, and group Lasso as special cases, for sparse high-dimensional misspecified Cox models with time-dependent covariates. Under suitable conditions on the true regression coefficients and random covariates, we provide oracle inequalities for prediction and estimation error based on the group sparsity of the true coefficient vector. The nonasymptotic oracle inequalities show that the penalized estimator has good sparse approximation of the true model and enables to select a few meaningful structure variables among the set of features.

Highlights

In recent years, high-throughput and nonparametric complex data have been frequently collected in gene-biology, signal processing, neuroscience, and other scientific fields
Blazere et al [5] study the properties of group Lasso estimator in sparse high-dimensional generalized linear models (GLMs) with group sparsity of the covariates, and the oracle inequalities for the prediction and estimation error
We provide unified nonasymptotic results in terms of oracle inequalities for prediction and estimation error, and this provides a theoretical justification for the consistency of weighted group Lasso estimator in Cox models

Summary

Introduction

High-throughput and nonparametric complex data have been frequently collected in gene-biology, signal processing, neuroscience, and other scientific fields. Blazere et al [5] study the properties of group Lasso estimator in sparse high-dimensional generalized linear models (GLMs) with group sparsity of the covariates, and the oracle inequalities for the prediction and estimation error. Zhou et al There have been considerable developments in oracle inequalities, not limited to the linear models and GLMs. Lemler [17] introduces a data-driven weighted Lasso to estimate Cox models by approximating the intensity (without using partial likelihood), and oracle inequalities in terms of an appropriate empirical K-L divergence are obtained. Honda and Hardle [11] consider group SCAD-type and the adaptive group Lasso estimator to do variable selection for Cox models with varying coefficients, and the L2 convergence rate is obtained for increasing-dimension setting p/n → 0

A brief review of Cox models

Key of nonasymptotic analysis

Sharp oracle inequalities from restricted eigenvalue conditions

Conclusions and future study