在這篇論文的第一部份，對於單變數區間刪減及截斷資
料情形下的存活時間分布，我們改寫Turnbull的演算法以
估計這種數據的分布函數，並提出四種無母數檢定，來判
斷兩個存活時間的母體是否來自相同的分布，再由模擬的
結果去比較所提出的統計量在不同分布情形下的檢定力。
此外，我們用所提出的方法分析一個AIDS的研究。

在第二部份中，關於雙變數區間刪減情形下的存活時間
分布，首先我們想要模擬出這種雙變數區間刪減的數據，
並定義兩個變數之間的相關程度。接著提出幾種無母數秩
檢定，來判斷雙變數區間刪減資料的兩個變數其分布是否
相互獨立並且是否相同，並用模擬的結果來顯示所提出的
統計量的檢定力。我們亦將所提出的方法應用到一個
AIDS的研究（ACTG 181）。

In Part 1 of this paper, we adapt Turnbull’s algorithm to estimate the distribution function of univariate interval-censored and truncated
failure time data. We also propose four non-parametric tests to test whether two groups of the data come from the same distribution. The
powers of proposed test statistics are compared by simulation under different distributions. The proposed tests are then used to analyze an AIDS study.

In Part 2, for bivariate interval-censored data, we propose some models of how to generate the data and several methods to measure the
correlation between the two variates. We also propose several nonparametric tests to determine whether the two variates are mutually independent or whether they have the same distribution. We demonstrate the performance of these tests by simulation and give an application to AIDS study（ACTG 181）.

PART 1. Univariate interval-censored and truncated data
1. Introduction……………………………………….………………….1
2. Adapted Turnbull’s algorithm…………………………….…………3
3. Urn model to select data………………………...……………………6
3.1. The selection of interval-censored data…………..…...…………..6
3.2. The selection of interval-censored and truncated data……..…....10
4.The proposed four non-parametric tests…………………..…...….13
4.1. The multinomial test…..…………….…………………………...14
4.2. The modified Sun’s rank test………...……….………………….16
4.3. The generalized weighted rank test……..….…………………....18
4.4. The generalized Mann-Whitney test……………..……...……….20
5. Simulation result.………………………….………………………...24
5.1. Exponential distribution…………………………...…………….26
5.2. Weibull distribution…………………………………..………….31
5.3. Extreme-value distribution…………………...………………….36
5.4. Logistic distribution………………...……………………………41
5.5. Conclusion of the simulation………………………………...…..46
6. Application to an AIDS study……..……………..…..……………..48
7. Reference of Part 1………………………………….………………51

PART 2. Bivariate interval-censored data
1. Introduction………………………..…………….…………...……..52
2. Generalized Turnbull’s algorithm……………...…………………..55
3. Selection of bivariate interval-censored data……..……...………..57
3.1. A 3×3 bivariate urn model……..…………...………..…………..57
3.2. A 4×3 bivariate urn model…...…..………………………..…..…67
3.3. A modified bivariate urn model……….…..……………80
3.4. A selection model from conditional probability density function…
…………………………....………..………………....…………90
4. Tests of independence for the two components of the bivariate
interval-censored data……….….…………….……….....…...…...95
4.1. Generalized Pearson’s correlation coefficient.…………….....….95
4.2. An extension of Spearman’s rho………..……...…………...……97
4.3. An extension of Kendall’s tau……………...….…..…..……….100
4.4. Generalized Wilcoxon signed rank test…………...……………103
4.5. Simulation result for independence……..……..………...……..106
5. Tests of whether the two components have the same distribution
in a bivariate interval-censored data……...…….……………….108
5.1. Tests under independence or not……….…..…………….…….108
5.2. Simulation result for identical distributions…….…….………..111
6. Application to an AIDS study（ACTG 181）……...……..……..113
7. Reference of Part 2………………………………..……….………117

Reference of Part 1
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survival data, with application to AIDS. Biometrics. Vol. 45, 1-11.
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AIDS studies. Biometrics. Vol. 51, 1096-1104.
[5] Sun, J. (1996)：A non-parametric test for interval-censored failure
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[6] Hung-Yen Hsu (1999)：The distribution of a non-parametric test for
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[8] Yu-Yu Kuo (2000)：A generalization of rank tests based on interval-
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Reference of Part 2
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