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

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

Received: 2 March 2004
Revised: 24 March 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 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 α at a point τ our main result states that 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.