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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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.

Article information

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

Dates
First available in Project Euclid: 5 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1346850066

Digital Object Identifier
doi:10.1214/12-AOS1015

Mathematical Reviews number (MathSciNet)
MR3015036

Zentralblatt MATH identifier
1257.62017

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

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

Citation

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. https://projecteuclid.org/euclid.aos/1346850066


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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.