Abstract

The linear spline model is able to accommodate nonlinear effects while allowing for an easy interpretation. It has significant applications in studying threshold effects and change-points. However, its application in practice has been limited by the lack of both rigorously studied and computationally convenient method for estimating knots. A key difficulty in estimating knots lies in the nondifferentiability. In this article, we study influence functions of regular and asymptotically linear estimators for linear spline models using the semiparametric theory. Based on the theoretical development, we propose a simple semismooth estimating equation approach to circumvent the nondifferentiability issue using modified derivatives, in contrast to the previous smoothing-based methods. Without relying on any smoothing parameters, the proposed method is computationally convenient. To further improve numerical stability, a two-step algorithm taking advantage of the analytic solution available when knots are known is developed to solve the proposed estimating equation. Consistency and asymptotic normality are rigorously derived using the empirical process theory. Simulation studies have shown that the two-step algorithm performs well in terms of both statistical and computational properties and improves over existing methods. Supplementary materials for this article are available online.

摘要

线性样条模型能够适应非线性效应，同时便于解释。它在研究阈值效应和变化点方面具有重要的应用。然而，由于缺乏经过严格研究和计算方便的节点估计方法，其在实践中的应用受到限制。估计节点的一个关键困难在于不可微性。在本文中，我们使用半参数理论研究线性样条模型的正则和渐近线性估计量的影响函数。基于理论发展，我们提出了一种简单的半光滑估计方程方法，以使用修改后的导数来规避不可微性问题，这与之前基于平滑的方法形成对比。在不依赖任何平滑参数的情况下，所提出的方法在计算上是方便的。为了进一步提高数值稳定性，开发了一种利用节点已知时可用的解析解的两步算法来求解所提出的估计方程。使用经验过程理论严格推导出一致性和渐近正态性。仿真研究表明，两步算法在统计和计算特性方面都表现良好，并且比现有方法有所改进。本文的补充材料可在线获取。使用经验过程理论严格推导出一致性和渐近正态性。仿真研究表明，两步算法在统计和计算特性方面都表现良好，并且比现有方法有所改进。本文的补充材料可在线获取。使用经验过程理论严格推导出一致性和渐近正态性。仿真研究表明，两步算法在统计和计算特性方面都表现良好，并且比现有方法有所改进。本文的补充材料可在线获取。