Kernel estimation of the copula function under twice censoring

We study the kernel estimation of the copula function of two twice censored random variables. Firstly, 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. To prove this convergence, we use the weak convergence of the estimator of the bivariate distribution function under twice censoring. This latter result has been established in a previous paper and it was based on a representation of the product limit estimator of the distribution function as a sum of centered, independent and identically distributed variables. This representation allowed to prove the requested weak convergence. Finally, we illustrate the performances of our estimators through a simulation study. The results of this study show that our estimators have a good quality of estimation.