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

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

doi:10.1214/12-AOS1015

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

Statistics Theory 2012-09-26 v3 Statistics Theory

Authors: 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.

Keywords

Grenander-type estimator L-infinity error limit distribution nonincreasing function supremum norm Gumbel distribution convergence rate

Cite

@article{arxiv.1111.5934,
  title  = {The limit distribution of the $L_{\infty}$-error of Grenander-type estimators},
  author = {Cécile Durot and Vladimir N. Kulikov and Hendrik P. Lopuhaä},
  journal= {arXiv preprint arXiv:1111.5934},
  year   = {2012}
}

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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)

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

Abstract

Keywords

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