Locally sparse quantile estimation for a partially functional interaction model

Area of Science:

• Statistics
• Functional Data Analysis
• Econometrics

Background:

• Functional data analysis is widely used.
• Existing models often assume simple error distributions and do not fully capture complex interactions.
• Partially functional models with scalar and functional covariates are gaining traction.

Purpose of the Study:

• To develop a novel partially functional model incorporating interactions between scalar and functional covariates.
• To address challenges posed by long-tailed error distributions and achieve interpretable estimation.
• To introduce a method that respects the main effect-interaction hierarchy and performs variable selection.

Main Methods:

• A partially functional model with linear scalar effects and nonlinear functional effects.
• Quantile regression for handling long-tailed error distributions.
• A penalization approach for estimation, local sparsity identification, and hierarchy adherence.
• Development of an effective computational algorithm.

Main Results:

• The proposed penalization approach effectively estimates model parameters and identifies local sparsity.
• Consistency properties of the estimation method are rigorously established under mild conditions.
• Simulation studies demonstrate the practical effectiveness of the approach.

Conclusions:

• The study presents a novel and practically useful partially functional model with interaction terms.
• The proposed estimation approach is statistically sound and numerically efficient.
• The method is applicable to real-world data, as shown by the Tecator data analysis.