Confidence Intervals for Sparse Penalized Regression with Random Designs.

Area of Science:

• Statistics
• Data Science
• Computational Statistics

Background:

• Sparse penalized regression is crucial for variable selection and estimation in large datasets.
• Existing methods for valid inference with general penalties (convex and non-convex) remain an active research area.
• The need for robust statistical inference methods in high-dimensional data analysis is growing.

Purpose of the Study:

• To propose a unified framework for constructing confidence intervals in sparse penalized regression.
• To address the challenge of valid inference for a wide range of penalty types, including convex and non-convex.
• To provide theoretical guarantees and empirical validation for the proposed inference procedure.

Main Methods:

• Utilizing advanced optimization tools based on stochastic variational inequality theory.
• Developing a general framework applicable to various sparse penalized regression models.
• Investigating inference for both population-level penalized regression parameters and underlying linear model parameters.

Main Results:

• A unified framework for constructing confidence intervals for sparse penalized regression is established.
• Theoretical convergence properties of the proposed inference method are rigorously derived.
• The proposed method demonstrates validity and effectiveness through simulations and real-world data analysis.

Conclusions:

• The developed framework offers a significant advancement in statistical inference for sparse penalized regression.
• The approach provides reliable confidence intervals across diverse penalty functions, enhancing data analysis capabilities.
• This work contributes to the robust statistical analysis of high-dimensional data, facilitating better insights.