線性迴歸模型在量測誤差與共線性影響下之估計方法初探

近幾年來，大數據之訊息分析與應用已經成為各領域中無所不在的重要技術；從學術研究到工商業發展，似有源源不絕的知識情報持續累積。在資料數據的分析研究中，有時可能需要較為複雜的統計方法(例如，沒有唯一解的參數估計式可以直接使用)，其中的部分原因也許可歸咎於測量的不精確性或無法預期的資料收集錯誤。此外迄今為止，已經有一些文獻回顧去說明如果共線性存在於數據結構中，統計分析可能導致嚴重錯誤的結論。因此，於資料分析和訊息掘取時，調查研究人員應該注意這種情況，並在解讀數據時慎選統計方法避免錯誤結論發生。本篇論文裡，我們研究了線性模型的量測誤差與共線性問題。作為解釋因果關係可能性的一種工具，迴歸分析已在悠久的歷史當中扮演關鍵的角色。本文的其中一個主要目的則是想要提醒讀者古典理論的模型估計程序(最小平方法)在某些應用上可能需要修正參數係數的估計偏差，此一偏差即來自於量測誤差或共線性之影響。根據上述，我們在這篇論文裡將提出兩個新的估計方法去修正可能的偏差。並提出可能適合進一步延伸的主題與研究觀點。這也是我們希望一些在本文所介紹的估計方法將有助於迴歸分析能夠處理這些偏差更一般性理論的後續發展。

關鍵字

線性迴歸；共線性；量測誤差

並列摘要

In recent years, big data applications and intelligence have become ubiquitous and keep discovering more insights from the academia and the industry/economy alike. Data analyses in those researches may sometimes require complex statistical methods where, in general, no closed-form solutions can be directly used because of imprecision of measurements and/or unexpected errors. In addition, there have been, to date, a number of literature reviews that some statistical results may lead to undesirable conclusions due to collinearity in data structure. Hence investigators should pay attention to such situations when digging the data and drawing the information. In this dissertation, we study the mis-measured and collinear issues in linear models. To be a tool for interpreting the possibility of cause-effect relations, regression analysis plays an important role for a long time. One of the main goals of this study is to remind the readership that classical estimation approach, least-squares method, may need to correct for the biases of parameter coefficients in certain applications. According to the aforementioned, we propose two new methods to correct such biases and give an outlook on further extended works. It is also our hope that some of the estimation approaches proposed in this dissertation will contribute to the subsequent development of a more general theory for biases correction in regression analysis.

並列關鍵字

Linear Regression ； Collinearity ； Measurement Error

參考文獻

Abarin, T. (2008). Second-order least squares estimation in regression models with application to measurement error problems. PhD Disseration, University of Manitoba.

Alheety, M. I., Kibria, B. M. G. (2012). Modified Liu-type estimator based on (r–k) class estimator. Communications in Statistics - Theory and Methods 42:304–319.

Apanasovich, T. V., Carroll, R. J., Maity, A. (2009). SIMEX and standard error estimation in semiparametric measurement error models. Electronic Journal of Statistics 3:318–348.

Aslam, M., Riaz, T., Altaf, S. (2013). Efficient estimation and robust inference of linear regression models in the presence of heteroscedastic errors and high leverage points. Communications in Statistics - Simulation and Computation 42:2223–2238.

Baltagi, B. H., Li, Q. (2002). On instrumental variable estimation of semiparametric dynamic panel data models. Economics Letters 76:1–9.

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線性迴歸模型在量測誤差與共線性影響下之估計方法初探

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