ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL

Area of Science:

• Statistics
• Biostatistics
• Machine Learning

Background:

• High-dimensional Cox proportional hazards models are crucial for survival analysis with many covariates.
• Sparse data and a large number of time-dependent covariates pose challenges for traditional statistical methods.
• Estimating regression coefficients accurately is vital for understanding risk factors and predicting outcomes.

Purpose of the Study:

• To develop and analyze an absolute penalized maximum partial likelihood estimator for sparse, high-dimensional Cox regression.
• To establish theoretical guarantees for the estimator's performance in settings where the number of covariates exceeds the sample size.
• To provide sharper oracle inequalities compared to existing methods.

Main Methods:

• Utilizing extensions of compatibility and cone invertibility factors of the Hessian matrix.
• Developing a novel approach based on natural extensions of these factors for time-dependent covariates.
• Proving oracle inequalities by demonstrating that key statistical quantities are bounded from below by positive constants.

Main Results:

• Established oracle inequalities for the penalized maximum partial likelihood estimator in high-dimensional sparse Cox models.
• Demonstrated that the compatibility and cone invertibility factors are bounded from below by positive constants, even with time-dependent covariates.
• Achieved sharper oracle inequalities compared to methods relying on restricted eigenvalues.

Conclusions:

• The proposed estimator offers improved theoretical performance in challenging high-dimensional survival data settings.
• The findings provide a robust theoretical foundation for using penalized likelihood methods in complex regression scenarios.
• This work advances statistical inference for time-dependent covariates in sparse, high-dimensional survival analysis.