In a bivariate setting, we consider the problem of detecting a sparse
contamination or mixture component, where the effect manifests itself as a
positive dependence between the variables, which are otherwise independent in
the main component. We first look at this problem in the context of a normal
mixture model. In essence, the situation reduces to a univariate setting where
the effect is a decrease in variance. In particular, a higher criticism test
based on the pairwise differences is shown to achieve the detection boundary
defined by the (oracle) likelihood ratio test. We then turn to a Gaussian
copula model where the marginal distributions are unknown. Standard invariance
considerations lead us to consider rank tests. In fact, a higher criticism test
based on the pairwise rank differences achieves the detection boundary in the
normal mixture model, although not in the very sparse regime. We do not know of
any rank test that has any power in that regime