MULTI-PARAMETRIC COPULA ESTIMATION BASED ON MOMENTS METHOD UNDER CENSORING

Abstract

This thesis combines two interesting branches of statistics: survival analysis and copula theory. The primary objective is to extend the copula theory results via semi-parametric estimation, under censored data. More precisely, we are interested by a copulas semi-parametric estimation, based on the classical moments estimation method, adapted for bivariate censored data. There are various kinds of censoring, we are only look at doubly and singly right-censored data. As theoretical results, general formulas were proved with analytical forms of the obtained estimators. According to early research, many asymptotic results obtained in the framework of non-parametric statistics for right-censored observations are based on the Kaplan Meier estimator, which estimates the survival function. Taking into account the results of Lopez and Saint-Pierre (2012) [72], Gribkova and Lopez (2015) [39], the asymptotic normality of the empirical survival copula was established for the two cases of censoring. The dependence structure between the bivariate survival times was modeled under the assumption that the underlying copula is Archimedean. Accounting for various censoring patterns (singly or doubly censored), a simulation study was performed efficiency and robustness of the new estimator proposed. Individual random parameters, which are commonly understood as frailty parameters, are another tool frequently employed for modeling multivariate survival data. We implemented this model for two-variable survival data using Archimedean copulas in the final part of the thesis. The frailty variables considered here are latent variables that are not observed, are nevertheless one-dimensional. In the example presented, this variable characterized the effect of the individual on the recurrence time. Then we looked at Clayton-Oakes copulas in particular, and even the model with gamma-type frailty. For each of these two models, the copulas used for the bivariate survival functions are the same. Even so, the marginal survival functions are modeled in different ways. The applications for health-related survival data were next examined.