线性样条指数回归模型：可解释性、非线性与降维
Linear spline index regression model: Interpretability, nonlinearity and dimension reduction
AI总结：
本文提出线性样条指数回归模型，兼具线性可解释性、非参数非线性拟合和单指数降维能力，并通过惩罚与卷积方法在节点数未知且可随样本量发散的情况下实现参数估计和节点识别。
摘要：
受某些现实世界数据集复杂性的启发，本文提出了一种新颖的灵活线性样条指数回归模型。该模型假设一个指数对响应变量具有分段线性效应，且变化在节点处连续发生。重要的是，该模型具备线性模型的可解释性，能够像非参数模型一样捕捉非线性效应，并实现类似于单指数模型的降维。此外，节点的位置和数量是未知的，这进一步增强了模型在实际应用中的适应性。我们提出了一种结合惩罚方法和卷积技术的新方法，以同时估计未知参数并确定节点数量。值得注意的是，所提出的方法允许节点数量随样本量发散。我们证明，所提出的估计量能够以趋近于1的概率识别节点数量，并且估计系数的效率与事先已知节点数量时一样高。我们还引入了一种检验节点存在性的程序。通过模拟研究和两个真实数据集来评估所提出方法的有限样本性能。
英文摘要：
Inspired by the complexity of certain real-world datasets, this article introduces a novel flexible linear spline index regression model. The model posits piecewise linear effects of an index on the response, with continuous changes occurring at knots. Significantly, it possesses the interpretability of linear models, captures nonlinear effects similar to nonparametric models, and achieves dimension reduction like single-index models. In addition, the locations and number of knots remain unknown, which further enhances the adaptability of the model in practical applications. We propose a new method that combines penalized approaches and convolution techniques to simultaneously estimate the unknown parameters and determine the number of knots. Noteworthy is that the proposed method allows the number of knots to diverge with the sample size. We demonstrate that the proposed estimators can identify the number of knots with a probability approaching one and estimate the coefficients as efficiently as if the number of knots is known in advance. We also introduce a procedure to test the presence of knots. Simulation studies and two real datasets are employed to assess the finite sample performance of the proposed method.
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