貝氏分析對迴歸曲線一致性之研究

The Consistency in Bayesian Regression Curve

蘇順國(Shun-Kuo Su)

指導教授 ： 吳裕振

中原大學/理學院/應用數學系/碩士(2008年)

https://doi.org/10.6840/CYCU.2008.00029

摘要

Chang, et al.在2007提出一種利用無母數貝氏估計來處理單調迴歸函數，其事前分佈是用伯氏多項式所引導出來的，而事後分配則是模仿Metropolis-Hasting algorithm (Green 1995)的可逆性所推導出來的。迴歸的問題在統計上是非常重要的，簡單的說是在研究自變數和應變數之間的關係，如：簡單的迴歸，更複雜的有曲線迴歸、曲面迴歸，都有很多人在研究（如：Gijbles 2004， Dette et al 2005等）也應用在其他方面，如：生物醫學統計、經濟學上等，其統計方法蠻多的，但我們認為必須要有理論來驗證其方法是對的。我們用一般圖形的限制迴歸來描述貝氏的架構，其事前分配是由伯氏多項式所提供，現在我們用貝氏估計大樣本性質的方法，其工具必須用Empirical processes的理論，和Kullback-Leibler divergence的概念來證明一致性，這裡面包含了單調回歸和一些其他圖形的限制迴歸，而一般圖形的限制迴歸包含單峰的迴歸、凹性的迴歸和S型迴歸等。我們也對迴歸曲線為階梯函數,Wen, et al.在2006年所提出的估計方法做探討，而他們並無大樣本性質，我們在此篇論文發現，只要再加上適當的條件，迴歸曲線為階梯函數的貝氏估計，其一致性的證明是類似的。

關鍵字

一般圖形的限制迴歸 ； 一致性 ； 貝氏分析 ； 階梯函數 ； 迴歸曲線

並列摘要

A nonparametric Bayesian approach by Chang, et al. (2007) was proposed to monotone regression where the prior is assumed by Bernstein polynomials and the posterior distribution is simulated by reversible jump Metropolis-Hasting algorithm (Green 1995)
The regression in the statistics is very important. Simply saying is to research the relation between the independent variable and the dependent variable. For example, the simple regression, curve regression and more complex curve regression. The approach studied by Gijbles (2004), Dette et al (2005) etc, which also could apply to the other fields, such as economics and biostatistics. It has many methods for statistics. But we have to confirm that the approach is credible.
We describe a Bayesian scheme for shape-restricted regression in which the prior is given by Bernstein polynomials. We present consistency theorems concerning the posterior distribution in this Bayesian approach. We must use the theory of Empirical processes and the Kullback-Leibler divergence to establish a consistency property of the Bayesian approach. This paper includes monotone regression and a few other shape-restricted regressions.
We study the Bayesian approach where the regression curve is step function was proposed by Wen, et al. (2006), but it has no large support. In this paper, we add some suitable condition. Accordingly, the establishment of consistency is similar.

並列關鍵字

Bayesian approach ； v-c class ； g-c class ； consistency

參考文獻

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7.CHANG, I. S., HSIUNG, C. A., WEN, C. C. & WU, Y. J. (2005). Bayesian survival analysis using Bernstein polynomials. Scand J. of Statist. 32, 447-466.