We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a k-monotone density g0 at a fixed point x0 when k>2. We find that the jth derivative of the estimators at x0 converges at the rate n−(k−j)/(2k+1) for j=0, …, k−1. The limiting distribution depends on an almost surely uniquely defined stochastic process Hk that stays above (below) the k-fold integral of Brownian motion plus a deterministic drift when k is even (odd). Both the MLE and LSE are known to be splines of degree k−1 with simple knots. Establishing the order of the random gap τn+−τn−, where τn± denote two successive knots, is a key ingredient of the proof of the main results. We show that this “gap problem” can be solved if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds.
Publié le : 2007-12-15
Classification:
Asymptotic distribution,
completely monotone,
convex,
Hermite interpolation,
inversion,
k-fold integral of Brownian motion,
least squares,
maximum likelihood,
minimax risk,
mixture models,
multiply monotone,
nonparametric estimation,
rates of convergence,
shape constraints,
splines,
62G05,
60G99,
60G15,
62E20
@article{1201012971,
author = {Balabdaoui, Fadoua and Wellner, Jon A.},
title = {Estimation of a k-monotone density: Limit distribution theory and the spline connection},
journal = {Ann. Statist.},
volume = {35},
number = {1},
year = {2007},
pages = { 2536-2564},
language = {en},
url = {http://dml.mathdoc.fr/item/1201012971}
}
Balabdaoui, Fadoua; Wellner, Jon A. Estimation of a k-monotone density: Limit distribution theory and the spline connection. Ann. Statist., Tome 35 (2007) no. 1, pp. 2536-2564. http://gdmltest.u-ga.fr/item/1201012971/