Abstract : The goal of this thesis is to study the local behavior of a probability measure, using a local regularity index. In rst part, we establish the asymptotic normality of the nearest neighbor density estimate and of the histogram. In the second one, we de- ne a mode estimator under weakened hypothesis. We show that the regularity index interferes in this two problems. Finally, we construct in third part various estimators of the regularity index. The thesis ends with a review on distribution function estimators.