區域多項式迴歸方法在斷點及折點問題上的研究

Abstract

區域線性迴歸方法在一般無母數迴歸問題（函數為二次微分連續的平滑曲線）上，有免於邊際效應的好性質。在斷點問題的研究上，我們想探討採用區域線性迴歸方法在邊際點及斷點附近的表現。另外對折點的問題，我們採用區域二次多項式迴歸方法估計折點及一次微分函數在折點上的跳躍大小值，並且導証出其漸近常態性，也在有限樣本的情況下以模擬來驗證這些估計量。

Consider the problem of estimating an unknown function that is smooth except for some change-points, where discontinuities occur on either the function or its first-order derivatives. Motivated by some appealing properties of local linear regression estimators, especially of no boundary effects, Shiau and Yeh (1995) proposed a jump-point estimator in the local linear regression context and suggested a back-fitting procedure to estimate the underlying regression function which has some jumps. This article investigates the boundary behaviors of the mean function estimator of Shiau and Yeh (1995), both in the boundary regions and neighborhoods of the jump points. An "optimal" rate for the bandwidths of the change-point estimator and the mean function estimator is derived. In regards to the cusp (or the change-point of the first derivative) estimation problem, we propose a cusppoint estimator based on maximizing the difference of two one-sided local quadratic estimaors and the corresponding jump size of the first derivative is the maximum difference. The asymptotic normality is established for both the cusp-point estimator and the jump size under some regular conditions. Finite sample properties are studied via simulations.