Date of Award
Doctor of Philosophy in Mathematical Sciences - (Ph.D.)
Ji Meng Loh
Many multivariate models have been proposed and developed to model high dimensional data when the dimension of a data set is greater than 2 (d ≥ 3). The existing multivariate models often force the “exchangeable” structure for part or the whole model, are not very flexible which tends to be of limited use in practice. There is a demand for developing and studying multivariate models with any pre-specified bivariate margins.
Suppose there exists such a class of flexible models with any pre-specified bivariate margins. Given a multivariate data, what is the distribution function and how to easily estimate the parameters from this multivariate model are often important issues to solve.
Dependent censoring has become an increasingly important issue in medical data analysis. Quite often failure times are subject to dependent censoring and how to model and quantify such dependence is also of great interest.
The research described in Chapter 2 of this dissertation has been motivated by the above challenging questions. Copula models are used to address these important problems.
The first result is to generalize the model construction approach proposed by Chakak (1993) to d?dimensional models with arbitrarily pre-specified bivariate margins. The second result is to give the distribution functions for models constructed using the construction approach proposed in the first result. The third result is to propose parameters estimation approach and new model selection approach for models constructed using the construction approach proposed in the first result. Simulation studies show that the parameter estimate works very well.
The research described in Chapter 3 of this dissertation has been motivated by the dependent censoring. The copula-graphic estimator (Zheng and Klein 1996) is first derived in this dissertation for marginal survival functions using Archimedean copula models based on semi-competing risks data. And its uniform consistency and asymptotic properties are proved.
A parameter estimation strategy is given to analyze the semi-competing risks data using Archimedean copula models. The method described in this dissertation is important and flexible in that it allows us to determine dependence levels between competing risks when two dependent competing risks are subject to independent censoring.
Based on the parameter estimation strategy proposed above, a new model selection procedure is given. An easy way to accommodate possible covariates in data analysis using the strategies is discussed.
Simulation studies show that the parameter estimate outperforms the estimator proposed by Lakhal, Rivest and Abdous (2008) for the Hougaard model and the model selection procedure works quite well. A leukemia data set is fitted by using the proposed model selection procedure and this dissertation end with some discussion.
Guo, Ziyan, "Asymmetric multivariate archimedean copula models and semi-competing risks data analysis" (2021). Dissertations. 1633.