Smoothed Quantile Regression with Large-Scale Inference.

Area of Science:

• Statistics
• Econometrics

Background:

• Quantile regression models heterogeneous effects between variables.
• Statistical inference in high-dimensional settings poses significant challenges.

Purpose of the Study:

• To develop a computationally efficient and statistically robust method for quantile regression inference under increasing dimensions.
• To introduce the 'conquer' algorithm and analyze its theoretical properties.

Main Methods:

• Convolution smoothing transforms the non-differentiable check function into a twice-differentiable, convex surrogate.
• Gradient-based optimization and multiplier bootstrap are employed for efficient computation and inference.
• Non-asymptotic bounds are derived for estimation and linearization errors.

Main Results:

• The 'conquer' method provides adequate approximation for computation and inference in quantile regression.
• Asymptotic normality is established under weaker dimensionality requirements than traditional methods.
• The multiplier bootstrap is validated for statistical inference.
• Numerical studies demonstrate the practicality and reliability of 'conquer' for large-scale inference.

Conclusions:

• 'conquer' offers a practical and scalable solution for high-dimensional quantile regression.
• The method facilitates robust statistical inference in complex data settings.
• An R package is available for implementing the 'conquer' methodology.