Bent-cable Regression Applied to the Daily Maximum Temperature in Cairo Through 2021

We use the so-called bent-cable model to describe the daily maximum temperature in Cairo, Egypt through 2021which 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 the recorded maximum temperature in Cairo to demonstrate the regularity and irregularity of bent-cable regression encountered in finite-sample settings. We also presented the projections of the bent-cable for 2022 compared to those of accuweather and broken stick projections. Practical conditions on the design are given to ensure regularity of the full bent-cable estimation problem, if the underlying bend segment has non-zero width. Under such conditions, the least-squares estimators are shown (i) to be consistent, and (ii) to asymptotically follow a multivariate normal distribution. Hence the least-squares estimators are used to analysis the considered data.