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 comparisons.

翻译：广义部分线性单指标可加模型因其在保持可解释性的同时整合函数灵活性与参数降维的通用性，已在多个领域得到广泛应用。然而，其估计过程面临严峻的计算挑战。本文提出一种新颖的稳定方法，该方法利用每个单指标效应（由其单指标系数定义）的模型矩阵以及惩罚完全Fisher信息矩阵，在统一迭代框架内动态更新单指标协变量的边界。导出的模型矩阵使得能够快速计算单指标效应的估计有效自由度与逐点置信带。通过广义Fellner-Schall方法将平滑参数更新整合到迭代过程中，该方法复用已导出的矩阵分解，从而为全局惩罚优化问题提供高效近似。在非高斯分布下中等样本量的模拟研究证实了该方法在多种场景下的估计一致性。值得注意的是，当现有先进竞争方法无法恢复真实单指标系数与非线性函数时，所提方法仍保持稳定，且在计算强度最大的场景下比常规两步法快80.13倍。通过Capital Bike Sharing数据的应用实例展示了建模优势——我们针对每年不同单指标系数处理单指标交互效应，这种复杂结构使竞争方法难以适用。所提方法已在R中实现，并提供函数以确保结果可复现与比较透明性。