ADAPTIVE ROBUST VARIABLE SELECTION.

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• Heavy-tailed high-dimensional data present significant challenges in statistical analysis.
• Existing methods like weighted robust Lasso (WR-Lasso) attempt to address these challenges using L1-penalty with weights.
• Bias issues from L1-penalty require further refinement for ultra-high dimensional settings.

Purpose of the Study:

• To investigate the model selection oracle property and asymptotic normality of WR-Lasso in ultra-high dimensions.
• To develop a practically feasible and theoretically sound method for handling heavy-tailed high-dimensional data.
• To propose an adaptive approach that ensures desirable asymptotic properties.

Main Methods:

• Utilizing penalized quantile regression with a weighted L1-penalty (WR-Lasso).
• Investigating theoretical properties including model selection oracle property and asymptotic normality under mild error distribution conditions.
• Proposing a two-step adaptive robust Lasso (AR-Lasso) procedure for practical implementation.

Main Results:

• Established theoretical guarantees for WR-Lasso, including oracle property and asymptotic normality.
• Demonstrated the necessity of adaptive weight vector selection for optimal performance.
• Validated the theoretical findings through numerical studies showcasing AR-Lasso's favorable finite-sample performance.

Conclusions:

• The proposed adaptive robust Lasso (AR-Lasso) effectively handles heavy-tailed, ultra-high dimensional data.
• AR-Lasso possesses desirable statistical properties like the oracle property and asymptotic normality.
• Numerical results confirm the practical utility and superior performance of AR-Lasso.