Non parametric estimation in the presence of associated and twice censored data

In this work, we are concerned with the non parametric estimation of some functional characteristics of a distribution of interest. We consider a context of association and twice censoring. Association represents a very interesting type of dependence given its large scope of applications as well as its ability to cover many useful random processes. Furthermore, in the context of twice censoring, the variable of interest X is right censored by a random variable R, min⁡(X, R) is itself left censored by a random variable L, and the three latent variables are independent. First, we establish the uniform almost complete convergence, with rate, of the product-limit estimator of the distribution function. Then, on the basis of this latter one, we propose a kernel estimator of the density function. This estimator is defined as a convolution between the product-limit estimator of the distribution function and an appropriately scaled kernel. We also establish the uniform almost complete convergence, with rate, of this estimator. Furthermore, we prove the rates of almost complete convergence for the kernel failure rate and mode estimators that result. Finally, we illustrate the performances of our studied estimators through a simulation study.