A Goodness-of-fit Test for Semi-parametric Copula Models of Right-Censored Bivariate Survival Times
According to Hougaard (2000), the fundamental difference between univariate and multivariate survival data is that assuming independence between multiple survival times is not appropriate. Thus, we have to specify the joint distribution of the survival times, which can be achieved by two steps. First, characterizing the properties of each survival time. Then, describing the joint behavior of survival times based on the properties of each survival time from the first step. Recently, copula model has become a popular choice of studying multivariate data. There is a major advantage of copula models, which is that the dependence structure can be studied separately from marginal distributions. In addition Archimedean copula is the most suitable copula family to describe right censoring multivariate survival data because they allow modeling dependence in arbitrarily high dimensions with only one parameter. However, it might exist misspecification when Archimedean model is not correct. Thus, checking the model specification is an important task in model diagnosis.
In this project, I first review the basic copula concept, construction methods of copulas and some commonly used copula families. Then we study the properties of survival copulas, especially Archimedean copula family, which includes some advantages of using copula model for multivariate data. In chapter 4, we will demonstrate the estimation of the parameter of the Archimedean model by using likelihood method, and then implement a goodness-of-fit test of Archimedean copula families on right censoring data. Goodness-of-fit test of Archimedean copula family on survival data has been developed for a decade history, which includes parametric and non-parametric ways. In chapter 5, we will present simulation studies and real data applications of our goodness-of-fit procedure. In the end, conclusion is provided.