Open Access
June 2012 The limit distribution of the $L_{\infty}$-error of Grenander-type estimators
Cécile Durot, Vladimir N. Kulikov, Hendrik P. Lopuhaä
Ann. Statist. 40(3): 1578-1608 (June 2012). DOI: 10.1214/12-AOS1015


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.


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Cécile Durot. Vladimir N. Kulikov. Hendrik P. Lopuhaä. "The limit distribution of the $L_{\infty}$-error of Grenander-type estimators." Ann. Statist. 40 (3) 1578 - 1608, June 2012.


Published: June 2012
First available in Project Euclid: 5 September 2012

zbMATH: 1257.62017
MathSciNet: MR3015036
Digital Object Identifier: 10.1214/12-AOS1015

Primary: 62E20 , 62G20
Secondary: 62G05 , 62G07

Keywords: extremal limit theorem , least concave majorant , monotone density , monotone failure rate , monotone regression , Supremum distance

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 3 • June 2012
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