臺灣博碩士論文加值系統

In this dissertation, we investigate the dependent relationship between two failure time variables which have a truncation relationship. Chaieb et al. (2006) considered semi-parametric framework under a “semi-survival” Archimedean-copula assumption and proposed estimating functions to estimate the association parameter, the truncation probability and the marginal functions.
In the first project, we adopt the same model assumption but propose different estimating methods. In particular we extend Clayton’s conditional likelihood approach (1978) to dependent truncation data for estimation of the association parameter. For marginal estimation, we propose a recursive algorithm and derive explicit formula to obtain the solution. The functional delta method is applied to establish large sample properties which can handle more general estimating functions than the U-statistic approach. Simulations are performed and the proposed methods are applied to the transfusion-related AIDS data for illustrative purposes.
Quasi-independence has been assumed by many inference methods for analyzing truncation data. By forming a series of tables, we also propose a weighted log-rank statisitcs for testing this assumption, which is our second project. Power improvement is possible by choosing an appropriate weight function. Here, we derive score tests when the dependence structure under the alternative hypothesis is specified semiparametrically. Asymptotic analysis and simulations are used to justify our proposed methods.

In this dissertation, we investigate the dependent relationship between two failure time variables which have a truncation relationship. Chaieb et al. (2006) considered semi-parametric framework under a “semi-survival” Archimedean-copula assumption and proposed estimating functions to estimate the association parameter, the truncation probability and the marginal functions.
In the first project, we adopt the same model assumption but propose different estimating methods. In particular we extend Clayton’s conditional likelihood approach (1978) to dependent truncation data for estimation of the association parameter. For marginal estimation, we propose a recursive algorithm and derive explicit formula to obtain the solution. The functional delta method is applied to establish large sample properties which can handle more general estimating functions than the U-statistic approach. Simulations are performed and the proposed methods are applied to the transfusion-related AIDS data for illustrative purposes.
Quasi-independence has been assumed by many inference methods for analyzing truncation data. By forming a series of tables, we also propose a weighted log-rank statisitcs for testing this assumption, which is our second project. Power improvement is possible by choosing an appropriate weight function. Here, we derive score tests when the dependence structure under the alternative hypothesis is specified semiparametrically. Asymptotic analysis and simulations are used to justify our proposed methods.

Chapter 1 Introduction 1
1.1 Motivation and Background 1
1.2 Overview of the Proposal 4
Chapter 2 Literature Review 6
2.1 Association Measures and Copula Models 6
2.2 Semi-parametric Inference for Survival-copula Models 8
2.3 Association Measures and Copula Models Suitable for Truncation Data 9
2.4 Statistical Inference for Truncated Data under Quasi-Independence 11
2.5 Statistical Inference for Dependent Truncated Data 13
Chapter 3 The Proposed Approach for Semi-parametric Inference 16
3.1 Estimation of Association 16
3.1.1 Conditional Likelihood Approach 16
3.1.2 Estimation based on Two-by-two Tables 17
3.1.3 Construction based on concordance indicators 18
3.1.4 Equivalent condition for different approaches 19
3.2 Estimation of marginal functions and truncation probability 20
3.2.1 The approach of Chaieb et al. (2006) 20
3.2.2. Recursive Solution to the Moment Constraints 21
3.3 Asymptotic Analysis 23
3.3.1 General Results for Asymptotic Properties 23
3.3.2. Asymptotic Behavior under Independence 24
3.4 Extension and Modification 30
3.4.1 Extension under right censoring 30
3.4.2 Modification for small risk sets 32
3.5 Numerical analysis 33
3.5.1 Simulation studies 33
3.5.2 Data analysis 37
3.6. Conclusion 39
Appendices : Project 1 40
Appendix 3.A: Asymptotic Analysis 40
Appendix 3.B: Equivalence of Different Estimating Functions 47
Appendix 3.C: Examples of AC Models 48
Chapter 4 Testing quasi-independence 51
4.1 The Proposed Test Statistics 52
4.1.1 Construction based on table 52
4.1.2 Relationship with Tsai’s test 54
4.2 Conditional Score Test 55
4.2.1 Likelihood Construction 55
4.2.2 Semi-survival AC models 57
4.3 Asymptotic analysis 60
4.3.1 Asymptotic normality 60
4.3.2 Variance estimation: Empirical vs. Jackknife 62
4.4 Modification for Right Censoring 63
4.4.1 The Weighted Log-rank Statistics under Censoring 63
4.4.2 The Conditional Score test under Censoring 65
4.4.3 Asymptotic Analysis under Censoring 65
4.5 Numerical Studies 67
4.5.1 Comparing Two Variance Estimators 67
4.5.2 Size of the Weighted Log-rank Test 68
4.5.3 Power of the Tests 70
4.6 Data Analysis 76
4.7 Conclusion 77
Appendices : Project 2 79
Appendix 4.A: Asymptotic Analysis 79
Appendix 4.B: Proof of Rquivalence Formula 87
Chapter 5 Future Work 90
References 91

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