Cahiers du CEREMADE

Unité Mixte de Recherche du C.N.R.S. N°7534
Abstract : Motivated by Hampel's birds migration problem, Groeneboom, Jongbloed and Wellner (2001b) established the asymptotic distribution theory for the nonparametric Least Squares and Maximum Likelihood estimators of a convex and decreasing density at a fixed positive point. However, estimation of the distribution function of the birds' resting times involves estimation of the first derivative of the true convex density at 0, a boundary point at which the estimators are not consistent. In this paper, we focus on the Least Squares estimator. Our goal is to show that consistent estimators of both the density and its first derivative at the origin can be based solely on the LSE. Following the idea of Kulikov and Lopuhaa (2006) in monotone estimation, we show that it suffices to take the LSE and its derivative at a sequence converging slowly to 0 at a rate alpha provided that alpha does not exceed 1/3. We establish the joint asymptotic distributions of the LSE and its derivative and show that alpha =1/5 should be taken as it yields the fastest rates of convergence.
Consistent estimation of a convex density at the origin: Back to Hampel's birds problem
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