Abstract:
The study of existing work shows that many of the asymptotic results obtained in the context of nonparametric statistics for right censored observations are based on the properties of the Kaplan Meier estimator of the survival function. So, since this estimator was generalized by Patilea and Rolin [2006] to the case of the twice censorship model, it became interesting to study the properties of the last estimator (the Patilea-Rolin estimator), this is the main purpose of this thesis. More precisely, we are interested in this type of censorship with strong mixing processes. In this framework, after deducing the law of the iterated logarithm for the Patilea-Rolin estimator, we show the uniform almost complete convergence of the distribution function estimators, with rate, first for the empirical distribution function based on α-mixing data. Then, in the case of left censorship, and under the same hypothesis of dependence, we specify the rate of this convergence for the estimator of the distribution function (which is deduced from that of Kaplan-Meier by inverting the time). We then exploit these two previous results to obtain the rate of the almost complete convergence of the Patilea-Rolin estimator as well as the kernel estimator of the cumulative failure rate, based on α- mixing data. To support our theoretical study, we present a simulation study accompanied by an application on real data. Starting from the result of Patilea and Rolin [2006], the kernel estimation of the density function for this model, was proposed by Kitouni et al. [2015]. Based on our previous results, we then continue the study of this last estimator under the condition of strong mixing. We establish its rate of the uniform almost complete convergence as well as that of the kernel failure rate estimator. It should be noted that the rates proposed in this thesis, under the condition of the strong mixing, are identical to those obtained for independent data.
