Estimation utilisant les polynômes de Bernstein
This thesis focuses on the presentation of the Bernstein estimators which are recent alternatives to conventional estimators of the distribution function and density. More precisely, we study their various properties and compare them with the empirical distribution function  and the kernel method estimators. We determine an asymptotic expression of the first two moments of the Bernstein estimator for the distribution function. As the conventional estimators, we show that this estimator satisfies the Chung-Smirnov property under conditions. We then show that the Bernstein estimator is better than the empirical distribution function in terms of mean squared error. We are interested in the asymptotic behavior of Bernstein estimators, for a suitable choice of the degree of the polynomial, we show that the Bernstein estimators are asymptotically normal. Numerical studies on some classical distributions confirm that the Bernstein estimators may be preferable to conventional estimators.
Non-parametric density estimator asymptotic properties empirical processes almost sure limits simulation study