A NEW PERSPECTIVE ON ROBUST

Area of Science:

• Statistics
• Robust Statistics
• Statistical Inference

Background:

• Least squares estimates are sensitive to heavy-tailed errors and outliers.
• Robust statistical methods are essential for reliable data analysis.

Purpose of the Study:

• To develop an adaptive Huber estimator robust to heavy-tailed distributions.
• To provide nonasymptotic concentration results for the adaptive Huber estimator.
• To apply these results to large-scale simultaneous inference and multiple testing.

Main Methods:

• Revisiting the Huber estimator with a sample-size-dependent tuning parameter.
• Deriving sub-Gaussian-type deviation inequalities and nonasymptotic Bahadur representations.
• Obtaining Berry-Esseen and Cramér-type moderate deviation results.
• Analyzing a dependence-adjusted multiple testing procedure.

Main Results:

• The adaptive Huber estimator demonstrates robustness against heavy-tailed errors.
• Nonasymptotic concentration results are established for finite second moments.
• Normal approximation results (Berry-Esseen, moderate deviation) are derived.
• The robust multiple testing procedure controls the false discovery proportion under mild conditions.

Conclusions:

• The adaptive Huber estimator offers a robust alternative to least squares for heavy-tailed data.
• The derived theoretical results support its application in complex statistical inference.
• The method shows promise for reliable analysis in large-scale simultaneous inference.