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.
"On combinatorial testing problems." Ann. Statist. 38 (5) 3063 - 3092, October 2010. https://doi.org/10.1214/10-AOS817