Contribution to the non parametric recursive estimation in relation with the extreme value index and the mixing hypothesis

Résumé

The main objective of this thesis resides in applying the stochastic approximation method to build up a large class of recursive non parametric kernel estimators for dependent and independent variables. First, we define a recursive kernel estimator of the conditional extreme value index. We investigate the properties of the proposed recursive estimator and compare it to Hill's non recursive kernel estimator. We show that using some optimal parameters, the proposed recursive estimator defined by the stochastic approximation algorithm proves to be very competitive to Hill's estimator. Efficiency and feasibility were confirmed by theoretical results and then by applications on simulated real data about Malaria in Senegalese children. Second, we extend the work of Slaoui (2014b) to the case of Alpha-mixing data. We study the properties of these estimators and compare them with Nadaraya's non recursive distribution estimator. Using an optimal choice of the bandwidth and an appropriate choice of the stepsize parameter, the recursive estimators allowed us to obtain quite better results compared to the non recursive distribution estimator under Alpha-mixing condition in terms of estimation error. We elaborate the central limit theorem and the uniform convergence for the proposed estimators under some mild conditions. The obtained theoretical results are corroborated through simulation study. Finally, we adopt the stochastic approximation algorithms to define a kernel estimator of the mode based on the recursive kernel density estimator developed by Mokkadem et al. (2009a). Additionally, we establish its almost sure convergence under strong mixing hypothesis and we confirm these theoretical results through numerical simulations.

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