Bent-Cable Regression Theory and Applications

Abstract

We use the so-called “bent-cable” model to describe natural phenomena that exhibit a potentially sharp change in slope. The model comprises two linear segments, joined smoothly by a quadratic bend. The class of bent cables includes, as a limiting case, the popular piecewise-linear model (with a sharp kink), otherwise known as the broken stick. Associated with bent-cable regression is the estimation of the bend-width parameter, through which the abruptness of the underlying transition may be assessed. We present worked examples and simulations to demonstrate the regularity and irregularity of bent-cable regression encountered in finite-sample settings. We also extend existing bent-cable asymptotics that previously were limited to the basic model with known linear slopes of 0 and 1. Practical conditions on the design are given to ensure regularity of the full bent-cable estimation problem if the underlying bend segment has nonzero width. Under such conditions, the least-squares estimators are shown to be consistent and to asymptotically follow a multivariate normal distribution. Furthermore, the deviance statistic (or the likelihood ratio statistic, if the random errors are normally distributed) is shown to have an asymptotic chi-squared distribution.