Generalized Functional Linear Models with Semiparametric Single-Index Interactions.

Area of Science:

• Statistics
• Econometrics
• Biostatistics

Background:

• Generalized linear models (GLMs) are widely used but often assume scalar predictors.
• Functional data analysis (FDA) extends GLMs to incorporate functional predictors, offering richer modeling capabilities.
• Existing functional GLM methods may lack parsimony or struggle with data-driven basis function selection.

Purpose of the Study:

• Introduce a novel class of functional generalized linear models (FGLMs) for scalar responses with functional covariates.
• Develop a parsimonious semiparametric modeling strategy using a single-index structure for interactions.
• Investigate estimation and asymptotic properties under different functional feature scenarios.

Main Methods:

• Proposed a two-step estimation procedure utilizing local estimating equations.
• Considered two scenarios: pre-determined basis functions (e.g., Fourier, wavelet) and data-driven basis functions (e.g., functional principal components).
• Developed theoretical asymptotic properties for parameter estimates.

Main Results:

• The proposed FGLM effectively models interactions between scalar covariates and functional predictors.
• Demonstrated that data-driven basis function estimation increases asymptotic variance of parameter estimates.
• Successfully applied the method to an empirical dataset, uncovering a significant, previously undetected interaction.

Conclusions:

• The novel FGLM provides a flexible and parsimonious framework for analyzing scalar responses with functional predictors.
• The findings highlight the trade-off between model flexibility and estimation precision when using data-driven functional features.
• The methodology offers a powerful tool for discovering complex relationships in functional data across various scientific domains.