伯氏多項式在部份單調迴歸之最大概似估計

Maximum Likelihood Estimator of Partial Monotone Regression Using Bernstein Polynomial

林慧欣(Hui-Hsin Lin)

指導教授 ： 吳裕振

中原大學/理學院/應用數學研究所/碩士(2011年)

https://doi.org/10.6840/CYCU.2011.00120

摘要

估計迴歸曲線是在研究其自變數和應變數的關係, 也能推測未來的走向, 讓我們
做出最佳決策的方法.
在本篇論文中, 主要是探討部份單調迴歸曲線, 並且加入共變數, 也就是可改變
其曲線之因子, 而且此曲線具有單調之性質. 我們採用伯氏多項式來描述我們的圖
形. 但也因為其函數比一般線性迴歸還要複雜許多, 而本篇論文是使用M.L.E. 去
做估計, 而且提供了一個來估計部份單調回歸曲線的方法.
在統計模擬方面, 我們去做不同樣本的資料模擬, 演算法用獨立演算法去計算且
軟體為R-程式. 我們不但完整的介紹出模型的建立及模擬結果, 並有不錯的表現.

關鍵字

伯氏多項式 ； 單調迴歸 ； 最大概似估計 ； 馬可夫鏈蒙地卡羅

並列摘要

Estimating the linear regression helps us to study the relationship
between the dependent variable and the independent variable, and we
can also predict the forecast which make us to find the best method.
The main point in the thesis is to probe the partial monotone linear
regression, and joins the covariance. That is, we may change the factor
of its curve and moreover this curve has the monotonous nature. We
use Bernstein polynomial to describe our graph. Because these functions
are more complex than other general linear regression functions, so we
estimate it by using M.L.E. We also provide a method to estimate the
partial monotone linear regression.
We simulated the sample with different numbers of data, and used independent
calculating method to calculate the conclusion with software-
R. We not only completely introduce model establishment and analogue
the result, but also compare with real data for our conclusion.

並列關鍵字

Bernstein Polynomial ； Monotone Regression ； Maximum Likelihood Estimator ； Markov Chain Monte Carlo

參考文獻

Department of Applied Mathematics, Chung Yuan Christian University, master thesis.

[1] 廖偉廷(2010), ”Bayesian partial monotone regression by using bernstein polynomial”.

[2] 王佐剛(2008), ”Bayesian regression with isotonic random berstein polynomials”.

restricted regression with random Bernstein polynomials”, Complex Dataset and

Inverse Problems. IMS Lecture Notes - Monograph Series , 54, 187-202.