ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, October 2005, Vol. 9, pp. 307-322
DOI: 10.1051/ps:2005014

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

Dietmar Ferger

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

(Received March 2, 2004. Revised March 24, 2005.)

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 F_n - F₀, where F₀ is another df which differs from F. If F and F₀ are locally Hölder-continuous of order $\alpha$ at a point $\tau$ 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 $\vert t\vert^{\alpha}$ .

Mathematics Subject Classification. 60E15, 60F05, 60F17, 62E20.

Key words: Rescaled empirical process, argmin-CMT, Poisson-process, weak convergence in $D(\mathbb{R} )$ .

© EDP Sciences, SMAI 2005