A new estimation method for continuous threshold expectile model

The Estimation of Bent Line Expectile Regression Model Based on a Smoothing Technique

“…Although the grid search method has become the mainstream estimation method of the change-point model, it has its limitations: the grid search method estimates the regression coefficients and change-point location separately; it requires a finer grid to identify the change-point location more precisely, which could result in expensive computation; it is not realistic for this method by assuming that the change-point only occurs at discrete grid points. In order to overcome these defects of the grid search method, Zhou and Zhang [21] extended the linearization technique proposed by Yan et al [22] to the bent line expectile regression model, converted the bent line expectile regression model into a standard linear expectile regression model, and then obtained the estimations of the regression coefficients and change-point parameter in the model (1) simultaneously through an iterative algorithm. The specific approach is to approximate the non-linear term by a firstorder Taylor expansion around an initial value 𝑡 ( ) ,…”

“…Although the estimation method proposed by Zhou and Zhang [21] makes up for the shortcomings of the grid search method, their method may also have a certain drawback, that is, estimation obtained by using the linearization technique in general may underestimate the change-point. Therefore, our next research content is to propose a better estimation method for the bent line expectile regression model, which can make up for the shortcomings of the grid search method and linearization method simultaneously.…”

“…In order to compare the proposed method with that of Zhou and Zhang [21], we also consider the linearization method. Tables A1-A6 show the simulation results of the linearization method ("seg") and the proposed method in this paper ("proposed") at the different expectile levels 𝜏 = 0.1,0.3,0.5,0.7,0.9.…”

“…Furthermore, it is not realistic for this method by assuming that the change-point only occurs at discrete grid points. Motivated by these issues, Zhou and Zhang [21] extended the linearization technique proposed by Yan et al [22] to the bent line expectile regression model and constructed the interval estimation of the estimators through the theory of the standard linear expectile regression model and the delta technique. Nonetheless, estimation obtained by using the linearization technique in general may underestimate the change-point.…”

“…Then the objective function is continuous and differentiable with respect to all parameters after being smoothed, so we can easily estimate all parameters in the model. This paper is to propose a new estimation method for the bent line expectile regression model through a smoothing technique, which can simultaneously compensate for the insufficiency of the grid search method of Zhang and Li [19] and the linearization method of Zhou and Zhang [21]. In addition, the theoretical proofs of the consistency and asymptotic normality of the proposed estimators are derived.…”