
Consistent estimation of a convex density at the origin
Balabdaoui, Fadoua (2007), Consistent estimation of a convex density at the origin, Mathematical Methods of Statistics, 16, 2, p. 77-95. http://dx.doi.org/10.3103/S1066530707020019
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Article accepté pour publication ou publiéDate
2007Journal name
Mathematical Methods of StatisticsVolume
16Number
2Publisher
Allerton Press, Inc.
Pages
77-95
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Balabdaoui, FadouaAbstract (EN)
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.Subjects / Keywords
Convex density; Hampel's problem; estimation at the boundary; Brownian motionRelated items
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