The limit distribution of the $L_{\infty }$-error of Grenander-type estimators

Abstract

Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n/\log n{)}^{-1/3}$ and the limiting distribution to be Gumbel.

Published in at http://dx.doi.org/10.1214/12-AOS1015 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Inconsistency of bootstrap: The Grenander estimator
• Bootstrapping the Grenander estimator
• On the ${\mathbb{L}}_p$-error of monotonicity constrained estimators
• Shape constrained nonparametric estimators of the baseline distribution in Cox proportional hazards model

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