The Limit Distribution of the Supremum-Error of Grenander-Type Estimators

• Activity Number: 691
• Type: Contributed
• Date/Time: Thursday, August 8, 2013 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Hendrik Lopuhaa*+ and Cecile Durot and Vladimir Kulikov
• Companies: Delft University of Technology and Univeristy of Nanterre and ASR Nederland
• Keywords: Supremum distance ; Extremal limit theorem ; Least concave majorant ; Monotone density ; Monotone regression ; Monotone failure rate

Abstract

Let $f$ be a non-increasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum distance between $f$ and its isotonic estimator on any interval $(\alpha_n, 1 - \alpha_n] \subset [0,1]$, where $\alpha_n$ tends to zero at a suitable rate. The rate of convergence of the supremum distance is found to be of order $(\log n /n)^{1/3}$ and the limiting distribution turns out to be Gumbel with a parameter depending on a functional of $f$ and $f'$. The results are obtained in a general framework, which includes the Grenander estimator of a decreasing density, the least squares estimator of a monotone regression curve or an isotonic estimator of a decreasing hazard of right-censored observations.