Smooth-threshold estimating equations for partially linear additive models based on modal regression

We study variable selection for partially linear additive models based on smooth-threshold estimating equations and modal regression. Employing B-spline basis functions to approximate the non-parametric component, we convert the semi-parametric model into a parametric model. Combining smooth-threshold estimating equations with modal regression, via appropriate bandwidth and regularization tuning parameters, we can select significant parametric variables and additive functions simultaneously, while acquiring their consistent estimators. One noteworthy benefit of this modal regression-based procedure is its robustness to outliers or heavy-tailed error distributions. Another highlight is the introduction of a smooth threshold estimating equations, which effectively avoids the challenge of non-convex optimization. Additionally, under certain mild conditions, we establish the theoretical properties. Simulation studies illustrate that the proposed method is effective in terms of estimation accuracy and variable selection, while practical applications further reinforce its excellent performance in real-world scenarios.