The limit distribution of the $L_{\infty}$-error of Grenander-type estimators
Cécile Durot, Vladimir N. Kulikov, Hendrik P. Lopuhaä
The Annals of Statistics · 2012 · Volume 40 · Issue 3
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.
Keywords
62E20, 62G05, 62G07, 62G20, extremal limit theorem, least concave majorant, monotone density, monotone failure rate, monotone regression, Supremum distance
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