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.






200722 

21052007 

Université
de PARIS  DAUPHINE Place du Maréchal de Lattre De Tassigny  75775 PARIS CEDEX 16  FRANCE Téléphone : +33 (0)1 44054923  fax : +33 (0)1 44054599 