Nonparametric Estimation of the Copula Function with Bivariate Twice Censored Data

Abstract

The aim of this thesis is to study the nonparametric estimation of the copula function in the presence of bivariate twice censored data. Assuming that the copula functions of the right and the left censoring variables are known, we propose an estimator of the joint distribution function of the variables of interest, then we derive an estimator of their copula function. Using a representation of the proposed estimator of the joint distribution function as a sum of independent and identically distributed variables, we establish the weak convergence of the empirical copula and simulation. After that, we studied the kernel estimation of the copula function of two twice censored random variables. So, we introduce two kernel estimators of the joint distribution function of the two variables of interest. Then, we use these estimators to propose two smoothed estimators of the copula function. We also prove the weak convergence of the proposed estimators to some tight centered Gaussian processes. Finally, we illustrate the performances of our estimators through a simulation study.