Variable selection for partially linear models via partial correlation.

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• Partially linear models (PLMs) are versatile semiparametric extensions of linear models.
• Existing methods for variable selection in PLMs with ultrahigh dimensional predictors are limited.
• Penalized least squares is a common but not universally applicable approach.

Purpose of the Study:

• To propose a novel variable selection procedure for partially linear models (PLMs) with ultrahigh dimensional predictors.
• To address limitations of existing penalized least squares methods.
• To provide a statistically rigorous method for identifying relevant predictors in complex models.

Main Methods:

• The proposed method utilizes partial correlation between partial residuals of the response and predictors.
• Theoretical properties, including model consistency, are systematically studied.
• Root-n convergence of coefficient estimators and asymptotic normality of baseline function estimates are established.

Main Results:

• The procedure demonstrates model consistency, ensuring correct variable selection in the limit.
• Theoretical convergence rates for coefficient estimators are proven.
• Asymptotic normality is established for the baseline function estimator, facilitating inference.

Conclusions:

• The developed variable selection procedure is theoretically sound and practically applicable for PLMs.
• The method offers advantages over existing techniques, particularly in ultrahigh dimensional settings.
• The approach is validated through Monte Carlo simulations and a real-world data example.