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arXiv · 1111.5934

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

Cécile Durot·Vladimir N. Kulikov·Hendrik P. Lopuhaä
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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.

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Cécile Durot, Vladimir N. Kulikov, Hendrik P. Lopuhaä. 2011-11-25. The limit distribution of the $L_{\infty}$-error of Grenander-type estimators. https://doi.org/10.1214/12-aos1015

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