ESAIM: Probability and Statistics

Table of Contents - November 2005 - Volume 9  

Research Article

On the minimizing point of the incorrectly centered empirical process and its limit distribution in nonregular experiments

Ferger, Dietmar

Department of Mathematics, Dresden University of Technology, Helmholtzstr. 10, 01062 Dresden, Germany; ferger@math.tu-dresden.de

Abstract

Let F n be the empirical distribution function (df) pertaining to independent random variables with continuous df F. We investigate the minimizing point $\hat\tau_n$ of the empirical process Fn - F0 , where F 0 is another df which differs from F. If F and F 0 are locally Hölder-continuous of order α at a point τ our main result states that $n^{1/\alpha}(\hat\tau_n - \tau)$ converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation and the drift-function are of the type |t|α.

(Received March 2 2004)

(Revised March 24 2005)

(Online publication November 15 2005)

Key Words:

  • Rescaled empirical process;
  • argmin-CMT;
  • Poisson-process;
  • weak convergence in $D(\mathbb{R})$ .

Mathematics Subject Classification:

  • 60E15;
  • 60F05;
  • 60F17;
  • 62E20
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