On combinatorial testing problems

Details On combinatorial testing problems On combinatorial testing problems

2009-08-24

arxiv.org/abs/0908.3437v2

Annals of Statistics 2010, Vol. 38, No. 5, 3063-3092

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/10-AOS817 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/10-AOS817

We study a class of hypothesis testing problems in which, upon observing the realization of an $n$-dimensional Gaussian vector, one has to decide whether the vector was drawn from a standard normal distribution or, alternatively, whether there is a subset of the components belonging to a certain given class of sets whose elements have been ``contaminated,'' that is, have a mean different from zero. We establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk. The combinatorial and geometric structure of the class of sets is shown to play a crucial role. The bounds are illustrated on various examples.

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