New approach for estimating the distribution tails for incomplete data

Abstract

our work is situated in the field of extreme values’ statistics for incomplete data namely the truncation and the censoring. In this context, several approaches for estimating distribution tails under random truncation have recently been developed: Gardes & Stupfler (2015) [18], Benchaira et al. (2015) [5], Benchaira et al. (2016a) [6], Benchaira et al. (2016b) [7], et Haouas et al. (2018) [21]. The first objective of this thesis is to define a new method ” the semiparametric method” to estimate the tail index of the distribution, while the majority of the existing method depend on the non-parametric estimator of the tail distribution index such as LyndeBell and Woodroofe, the ours is based on the semi-parametric estimator defined in Wang 1989 [48] that allows us introducing new estimators with high efficiency. For the second objective, at this point, we are interested in correcting the error of kernel estimators, such as Benchaira et al. (2016b)’s estimator, so we have introduced a new kernel estimator with reduced bias at the same time. Without forgetting the complete data, in the third objective of this thesis we add a new estimator of the extreme value’s index beside the well-known estimators such as Hill, Peng, ... etc. The new one is characterized by its robustness and stability and was developed by using the idea which was presented in Basu 1998 [2] based on the density power divergence function