In multivariate survival analyses, understanding and quantifying the association between survival times is of importance. Copulas, such as Archimedean copulas and Gaussian copulas, provide a flexible approach of modeling and estimating the dependence structure among survival times separately from the marginal distributions (Sklar, 1959). However, misspecification in the parametric form of the copula function will directly lead to incor- rect estimation of the joint distribution of the bivariate survival times and other model-based quantities.The objectives of this project are two-folded. First, I reviewed the basic definitions and properties of commonly used survival copula models. In this project, I focused on semi- parametric copula models where the marginal distributions are unspecified but the copula function belongs to a parametric copula family. Various estimation procedures of the de- pendence parameter associated with the copula function were also reviewed. Secondly, I extended the pseudo in-and-out-of-sample (PIOS) likelihood ratio test proposed in Zhang et al. (2016) to testing the semi-parametric copula models for right-censored bivariate sur- vival times. The PIOS test is constructed by comparing two forms of pseudo likelihoods, one is the "in-sample" pseudo likelihood, which is the full pseudo likelihood, and the other is the "out-of-sample" pseudo likelihood, which is a cross-validated pseudo likelihood by the means of jacknife. The finite sample performance of the PIOS test was investigated via a simulation study. In addition, two real data examples were analyzed for illustrative purpose.
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