臺灣博碩士論文加值系統

本篇論文主要研究現狀數據(current status data)的存活分析。 我們使用伯氏多項式(Bernstein polynomials)來當作累積風險函數的模型 , 利用馬可夫鏈蒙地卡羅之方法(Markov Chains Monte Carlo methods) , 簡稱 M.C.M.C. 法, 去估計累積風險函數的最大概似估計。 我們選擇伯氏多項式為模型 , 因為伯氏多項式易於考慮幾何資訊而且是平滑的連續函數 , 如:圖形為凹性(concave)或對累積風險函數的最初猜測。 本篇論文 , 我們用馬可夫鏈蒙地卡羅(M.C.M.C.)之方法去估計最大概似估計(Maximum Likelihood Estimator , M.L.E.)。
存活分析 , 是現在在統計上一個重要的研究 , 可被應用在醫學上 , 去計算存活率和平均存活時間 , 也可被保險業拿來估計保單的重要資訊 , 因此我們想了解它、理解它、研究它。 利用貝氏方法研究存活分析的有 Sinha & Dey (1997, 1998), 以及 Ibrahim et al. (2001) , 且用階梯函數估計右設限資料累積風險函數的有 McKeague & Tighiouart (2000, 2002)。 本篇論文 , 我們用伯氏多項式來估計現狀數據之累積風險函數 , 並且用 M.C.M.C. 之方法求 M.L.E.。
本篇論文的組織如下 : 第二節介紹 Bernstein 多項式的圖形與係數關係 , 並且由命題二去討論其反敘述的理論。 第三節介紹模型及其概似函數。 第四節將介紹演算法 : M-H Green 法 (Metropolis-Hastings Green Method)。 第五節為模擬計算 , 將伯氏多項式 M.L.E. 之估計的結果與 Step M.L.E. 做比較。 第六節則為結論與建議。

Survival analysis of current status data is studied using Bernstein polynomials Estimation Cumulate hazard functions Markov chain Monte Carlo (M.C.M.C.) methods estimation Maximum Likelihood Estimator (M.L.E.). These Bernstein polynomials easily take into consideration geometric information like concave or initial guess on the cumulative hazard functions , select only smooth functions ,
can have large enough support , and can be easily specified and generated. We use these M.C.M.C. methods Estimation MLE are quite satisfactory.
Survival analysis is a important research in the statistics. It is applied to calculate the survival rate and the mean survival time in the medicine , also estimate important information of insurance , so we want to study it. To study survival analysis using Bayes method like Sinha & Dey (1997, 1998), as well as Ibrahim et al. (2001).
Estimating cumulate hazard functions with the step functions like McKeague & right; Tighiouart (2000, 2002). To estimate cumulate hazard functions of current status data using Bernstein polynomials and find M.L.E. with M.C.M.C. methods in this paper.
This paper be organized as follows : Chapter 2 introduce the relations between polynomial coefficients and graphic structures , we discuss some problems about counter statements. Chapter 3 derive model and the likelihood function. Chapter 4 introduce algorithm : Metropolis-Hastings Green method. Chapter 5 is simulation study , we will compare Bernstein polynomials M.L.E. with Step M.L.E.. Chapter 6 is the conclusion and suggestion.

目錄
中文摘要 Ⅰ
英文摘要 Ⅲ
誌謝辭 Ⅴ
目錄 Ⅵ
圖目錄 Ⅶ
表目錄 Ⅷ
1 介紹 1
2 Bernstein 多項式之幾何性質 3
3 模型及其概似函數 4
3.1 關於Bernstein多項式之存活模型…………………………………4
4 參數估計…………………………………………………………………7
4.1 Metropolis-Hastings Green 演算法…………………………8
5 模擬計算…………………………………………………………………10
6 結論與建議………………………………………………………………14
參考文獻……………………………………………………………………15

圖目錄
1 樣本數為100的累積風險函數之圖形…………………………………11
2 圖1的累積風險函數轉換成分布函數之圖形…………………………11
3 樣本數為200的累積風險函數之圖形…………………………………12
4 圖3的累積風險函數轉換成分布函數之圖形…………………………12
5 樣本數為400的累積風險函數之圖形…………………………………13
6 圖5的累積風險函數轉換成分布函數之圖形…………………………13

表目錄
1 模擬結果………………………………………………………………10

Chang, I. S., Hsiung, C. A., WU, Y. J., and Yang, C. C.& (2005).
Bayesian survival analysis using Bernstein polynomials. Scand. J.
Statist. 32, 447-466.

Green, P. G. (1995). Reversible jump Markov chain Monte Carlo
computation and Bayesian model determination. Biometrika 82,
711-732.

Ibrahim, J. G., Chen, M. H. & Sinha D. (2001). Bayesian survival
analysis. Springer-Verlag, New York.

McKeague, I. W. & Tighiouart, M. (2000). Bayesian estimators for
conditional hazard functions. Biometrics 56, 1007-1015.

McKeague, I. W. & Tighiouart, M. (2002). Nonparametric Bayes
estimators for hazard functions based on right censored data.
Tamkang J. Math. 33, 173-189.

Sinha, D. & Dey, D. K. (1997). Semiparametric Bayesian analysis of
survival data. J. Amer. Statist. Assoc. 92, 1195-1212.

Sinha, D. & Dey, D. K. (1998). Surival analysis using semiparametric
Bayesian methods. In Practical nonparametric and semiparametric
Bayesian statistics (Edited by D. K. Dey, P. M¨uller & D.
Sinha), 195-211. Springer-Verlag, New York.

WU, Y. J. & Fang, W. Q. (2007). Bayesian survival analysis for current
status data, Master thesis. Chung Yuan Christian University.