Résumé en anglais

Motivated by Hampel's birds migration problem, \mycite{gjw:01b} established the asymptotic distribution theory for the nonparametric Least Squares and Maximum Likelihood estimators of a convex and decreasing density, $g_0$, at a fixed point $t_0 > 0$. However, estimation of the distribution function of the birds' resting times involves estimation of $g'_0$ at 0, a boundary point at which the estimators are not consistent. In this paper, we focus on the Least Squares estimator, $\tilde{g}_n$. Our goal is to show that consistent estimators of both $g_0(0)$ and $g'_0(0)$ can be based solely on $\tilde{g}_n$. Following the idea of \mycite{kuliandlopuh:06} in monotone estimation, we show that it suffices to take $\tilde{g}_n(n^{-\alpha})$ and $\tilde{g}'_n(n^{-\alpha})$, with $\alpha \in (0,1/3)$. We establish their joint asymptotic distributions and show that $\alpha =1/5$ should be taken as it yields the fastest rates of convergence.