Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

Hella Timmermann
Universität zu Köln
On sequential detection of a gradual change
DMV Tagung 2011 - Köln, 19. - 22. September
We will describe and analyze some sequential monitoring procedures for detecting a gradual change in
the drift parameter of a general stochastic process satisfying a certain (weak) invariance principle. It is
shown that the tests can be constructed such that the false alarm rate attains a prescribed level and
that the tests have asymptotic power one. A more precise analysis of the procedures under the alternative
proves that the stopping times, suitably normalized, have a standard normal limit distribution. A few results
from a small simulation study are also presented in order to give an idea of the finite sample behavior of
the suggested procedures.
Literatur
Steinebach, J. and Timmermann, H. (2011). Sequential testing of gradual changes in the drift of a stochastic
process. Journal of Statistical Planning and Inference, 141, 2682-2699.
Martin Wendler
Ruhr-Universität Bochum
Strong invariance principle for the generalized quantile process under dependence
A strong invariance principle for the empirical distribution function (the approximation by a Kiefer-Müller
process) has been established by Berkes and Philipp (1977) for dependent data. We extend this results to
the empirical U-process (the empirical process of the values h(Xi,Xi) for a bivariate, symmetric function
h). With the help of a generalized Bahadur representation, it follows that such a strong invariance principle
also holds for the empirical U-quantile process and consequently for GL-statistics (linear combination of
U-quantiles). GL-statistics have applications in robust estimation and robust change point detection. We
obtain the functional central limit theorem and the functional law of the iterated logarithm for GL-statistics
under dependence as straightforward corollaries.
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