We study a class of hypothesis testing problems in which, upon observing the realization of an ndimensional 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.
In REF , for various combinatorial and geometric structures of anomalous objects, conditions were established under which testing is possible or hopeless with a small risk.
On combinatorial testing problems
Annals of Statistics 2010, Vol. 38, No. 5, 3063-3092
Mathematics Computer Science