Stable direct estimation for GPLSIAMs using P-splines with dynamically updated boundaries

2026-05-20

AI Summary

This study addresses computational instability and high complexity in generalized partially linear single-index additive models (GPLSIAMs) under non-Gaussian distributions and complex structures. The authors propose a stable and efficient direct estimation method that, within a unified iterative framework, dynamically updates the boundary of single-index covariates using the model matrix and a penalized full Fisher information matrix, while simultaneously optimizing smoothing parameters via a generalized Fellner–Schall approach. This novel integration of dynamic boundary adjustment and joint smoothing parameter optimization substantially enhances estimation stability and computational efficiency, enabling flexible modeling of complex single-index interactions. Simulations demonstrate consistent and robust performance across various non-Gaussian settings, with computational speedups up to 80.13-fold. Applied to the Capital Bike Sharing dataset, the method successfully captures annual-varying single-index interaction effects that existing approaches fail to estimate.

Abstract

Generalized partially linear single-index additive models (GPLSIAMs) have been increasingly applied across diverse areas due to their versatility in integrating functional flexibility with parametric dimension reduction while maintaining interpretability. However, the estimation presents severe computational challenges. This paper introduces a novel stable method that uses the model matrix for each single-index effect, defined by its single-index coefficients, and the penalized complete Fisher information matrix to dynamically update the boundaries of the single-index covariates within a unified iterative framework. The derived model matrices enable the fast computation of the estimated effective degrees of freedom and pointwise confidence bands for the single-index effects. The smoothing parameter updates are integrated into the iterative process via the generalized Fellner-Schall method, which recycles the derived matrix decompositions, thereby providing an efficient approximation to the global penalized optimization problem. Simulation studies with moderate sample sizes under non-Gaussian distributions confirm the empirical consistency of the estimation across multiple scenarios. Notably, the proposed approach remains stable where state-of-the-art competitive methods fail to recover true single-index coefficients and nonlinear functions, and is 80.13 times faster than the usual two-step method in the most computationally intensive scenario. The modeling advantage is illustrated through an application to Capital Bike Sharing data, where we deal with a single-index interaction effect for each year, with distinct single-index coefficients, a complex structure that makes competitive methods inapplicable. The proposed method is implemented in R, with functions available for reproducibility and transparency in the comparisons.

Problem

• GPLSIAMs
• computational challenges
• single-index estimation
• non-Gaussian distributions
• model stability

Innovation

• GPLSIAMs
• P-splines
• dynamic boundary updating
• generalized Fellner-Schall method
• penalized Fisher information