臺灣博碩士論文加值系統

在本論文中主要包含兩個主題，一為具有相同進入時間的 Cox 迴歸模型之多維參數的重複顯著性檢定 (repeated significance test)，
另一為常態分配平均數的貝氏序列區間估計(Bayes sequential interval estimation)
。
在進行存活分析時序列分析是一種經常使用的方法，對具有參數的存活資料(parametric survival data)進行參數檢定時，Chang, Hsiung 和 Hwang (1999) 對於具有參數和不同進入時間的存活資料，利用鞅論(martingale theory)及計數過程 (counting process) 的方法，提出了一個能同時處理多維參數的重複顯著性檢定 (repeated significance test)。本文將利用他們所提出的方法針對具有相同進入時間的 Cox 迴歸模型，提出一個可檢定多維參數的重複顯著性檢定。為了簡單起見，我們只考慮二維參數的情形。由於 Cox 模型的部分概似分數過程 (partial likelihood score process) 並不是正交鞅 (orthogonal martingale)，故直接利用它們來做序列檢定並不適當，因此我們利用此部分概似分數過程去架構出一組正交鞅，並研究這組正交鞅的漸近性質，經過適當的隨機時間轉換(random time change)，它們會漸近於二維的布朗運動(Brownian motion)，而且其分量間彼此是獨立的。再利用 Siegmund(1985) IV.2 節布朗運動之重複顯著性檢定中的結果，就可提出一個用以檢定二維參數的重複顯著性檢定了。最後我們分別利用數值模擬和分析實際存活資料的方式，來驗証我們所提出的重複顯著性檢定的確是可行的。
有關貝氏序列區間估計的問題主要是要去找出一組最優的序列區間估計程序，它包含了一個最優的停止法則和一個最優的區間估計。在過去有關貝氏序列的區間估計的文獻,如 Blumenthal (1970)、Gleser 和 Kunte (1976) 等，他們所提出的貝氏序列區間估計程序都與事先分配有關。然而 Hwang (1999) 在對參數進行點估計時，提出了一個與事先分配無關的貝氏序列估計程序，我們延續此一概念，將其用於處理有關區間估計的問題。在本文中我們針對變異數已知的常態分配平均數，提出一個與事先分配無關的貝氏序列區間估計程序，而且其具有已確定的停止法則。
在最後我們也証明了此貝氏區間估計程序具有 Bickel 和 Yahav (1967,1868) 所定義之漸近點最優 (asymptoticlly pointwise optimal)
和漸近較佳 (asymptotically optimal)的性質。

Two subjects are considered in this paper. The first subject is repeated significance tests for multi-dimensional parameter in the Cox regression model with the simultaneous entry. The second subject is the Bayes sequential interval estimation of the mean of normal distribution.
The sequential method has been used extensively in survival analysis.Sometimes, we are interested to test the multi-dimensional parameter in the survival model. For parametric survival data with staggered entry, Chang, Hsiung and Hwang (1999) proposed a martingale method of constructing repeated significance tests to test a multi-dimensional parameter. Here we apply their method to the case of Cox regression model with
simultaneous entry. By the martingale central limit theorem of
Rebolledo (1980), we study the asymptotic properties of the orthogonal martingale which is constructed by the partial likelihood score process. Through making a random time change for the orthogonal martingale, we apply the strong representation theorem to obtain that the orthogonal martingale converges weakly to a standard $R^2$-valued Brownian motion.
With the result of repeated significance tests for Brownian motion (cf. Siegmund (1985) IV.2, pp. 73-81), we propose repeated significance tests for multi-dimensional parameter. A simulation study and a survival data study are included to indicate our tests are satisfactory.
Next, the problem of Bayes sequential interval estimation of the mean of a normal distribution with known variance is considered. The Bayes sequential interval estimation problem is to seek an optimal interval estimation procedure which includes an optimal stopping rule and an optimal interval estimate. In the past such researchers as Blumenthal (1970) and Gleser and Kunte (1976) proposed the Bayes sequential interval estimation procedure for some parameter, but their procedures depend on the prior distribution. Not depending on the prior distribution, an interval estimation procedure with deterministic stopping rule is proposed here. It is shown that the proposed procedure is asymptotically pointwise optimal and asymptotically Bayes in the sense of Bickel and Yahav (1967, 1968) for a large class of prior distributions.

目錄 i
中文摘要 ii
英文摘要 iv
第一章 緒論
第1.1節 背景及研究動機 1
第1.2節 本文架構 3
第二章 Cox 迴歸模型中多維參數的重複顯著性檢定
第2.1節 簡介 4
第2.2節 Cox 迴歸參數的重複顯著性檢定 6
第2.3節 數值模擬研究 22
2.3.1 完全序列檢定 22
2.3.2 群組序列檢定 28
第2.4節 實例分析 30
第三章 常態分配平均數的貝氏序列區間估計
第3.1節 簡介 34
第3.2節 漸近點最優與漸近較佳 36
第3.3節 貝氏序列區間估計程序 38
第四章 結論 44
附錄
A1. 算式的推導 45
A2. 本文所引用的相關定義及定理 52
A3. 罹患黑色素瘤的病人之存活資料 59
參考文獻 64

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