具有AR(1)誤差的迴歸模型的穩健性估計

Abstract

我們主要討論具有AR(1)誤差的線性及非線性迴歸模型的穩健性估計問題。在線性迴歸部份，對於相關參數我們提出了廣義中位數估計、feasible廣義中位數估計、廣義修正平均值估計以及feasible廣義修正平均值估計。在定義估計之前，我們運用誤差項的變異數矩陣對迴歸模型作Cochran-Orcutt轉換。轉換後的模型為一般線性迴歸模型，其誤差項為獨立且具相同分配的隨機變數。

廣義中位數估計及feasible廣義中位數估計，由於採用中位數的方法，所以有穩健性的特性。廣義修正平均值估計以及feasible廣義修正平均值估計，則是藉由Koenker與Bassett 在1978年提出的 regression quantile對資料先作部份刪除再作估計，所以對異常值不敏感，也具有穩健性的特性。

在非線性迴歸模型部份我們延伸廣義中位數估計的定義。在此論文中，除了定義穩健性估計之外，也討論Bahadur表示式、大樣本理論以及相關的資料模擬與應用。

In this thesis we focus on the linear and non-linear regression with AR(1) error models. In the linear regression part, the generalized median estimator, feasible generalized median estimator, generalized trimmed mean (GTM) and the feasible generalized trimmed mean (FGTM) are proposed. Before defining the estimators, we use the covariance matrix of the error terms to do a Cochran-Orcutt transformation on the regression model such that the transformed one is a usual linear regression model with i.i.d. error variables. Then we discuss the robust estimators of this transformed model.

The generalized and feasible generalized median estimators are defined by the l1 norm method. So they are robust to outliers. For the generalized and feasible generalized trimmed mean estimators, we apply sample regression quantile which is defined by Koenker and Bassett (1978) to trim data first and then define the estimators based on the rest of data. Due to trimming, these estimators are robust to outliers also. Besides the linear regression with AR(1) error model, we extended the idea of trimmed mean to introduce generalized and feasible generalized trimmed means for the nonlinear regression with AR(1) error model.

The corresponding Bahadur representations and asymptotic normality are proved in this thesis also. Besides these theoretical results, we also do simulations to discuss the effect of the estimation of correlation coefficients ρ to the model. And an application on a real data set is also given.