The Annals of Statistics

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

Cécile Durot, Vladimir N. Kulikov, and Hendrik P. Lopuhaä

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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.

Article information

Ann. Statist., Volume 40, Number 3 (2012), 1578-1608.

First available in Project Euclid: 5 September 2012

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Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62G20: Asymptotic properties
Secondary: 62G05: Estimation 62G07: Density estimation

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


Durot, Cécile; Kulikov, Vladimir N.; Lopuhaä, Hendrik P. The limit distribution of the $L_{\infty}$-error of Grenander-type estimators. Ann. Statist. 40 (2012), no. 3, 1578--1608. doi:10.1214/12-AOS1015.

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Supplemental materials

  • Supplementary material: Supplement to “The limit distribution of the $L_{\infty}$-error of Grenander-type estimators”. Supplement A: The supremum of the limiting process. Supplement B: Preliminary results for the inverse process. Supplement C: Points of jump.