Estimation of Knots in Linear Spline Models

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.

Keywords:

• Change-Point
• Differentiable in quadratic mean
• Efficiency
• Gradient-descent
• Newton–Raphson
• Semiparametric semismooth estimating equation

Supplementary Materials

The online supplementary materials contain proofs for Lemmas 1-3, Proposition 1, Theorems 1 and 2, the detailed algorithm for the single-knot situation, additional results regarding the gradient descent algorithm, and the code for the proposed method.

Acknowledgments

The author thanks to the anonymous reviewers and associate editor for the constructive inputs.