區域多項式迴歸平滑方法應用在改變點問題上之研究

Abstract

本文採用區域多項式迴歸平滑方法來估計函數的改變點的位置及其改變 量. 我們導証出此二估計量之漸近常態性, 並比較多項式次數的不同對改 變量之期望值及變異數所造成的影響.我們也探討了採用區域多項式迴歸 平滑方法在邊際點的表現, 並發現存在改變點效應.此外我們也提出了驗 証改變點階數的方法, 並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 expect for some change-points, where discontinuities occur on either the function or its derivatives. In this paper, we propose estimators for the location and jumpsize of the discontinuity, respectively, based on one-sided local polynomial regression smoothers. The asymptotic normality is established for both the change-point and jump size estimators under regularily conditions. Estimators of the mean function and its derivatives are also proposed. The boundary behaviors of these estimators are investigated, including the boundary regions and neighborhoodsof the change-point. It is found that the resulting estimators are free of the boundary effects. Unfortunately, there is a change-point effect due to the errorsfrom the estimation of the location and the jump size of the change- point. In addition, we give some theoretical reasons to distinguish cases between p-nu oddand p-nu even, where p is the order of the local polynomial and nu is the order of the discontinuities of the fuction at the change-point. Finite sample properties are studied via simulations.