A test of location for exchangeable multivariate normal data with unknown correlation

We consider the problem of testing whether the common mean of a single nn-vector of multivariate normal random variables with known variance and unknown common correlation ρρ is zero. We derive the standardized likelihood ratio test for known ρρ and explore different ways of proceeding with ρρ unknown. We evaluate the performance of the standardized statistic where ρρ is replaced with an estimate of ρρ and determine the critical value cncn that controls the type I error rate for the least favorable ρρ in [0,1]. The constant cncn increases with nn and this procedure has pathological behavior if ρρ depends on nn and ρnρn converges to zero at a certain rate. As an alternate approach, we replace ρρ with the upper limit of a (1−βn)(1−βn) confidence interval chosen so that cn=ccn=c for all nn. We determine βnβn so that the type I error rate is exactly controlled for all ρρ in [0,1]. We also investigate a simpler approach where we bound the type I error rate. The former method performs well for all nn while the less powerful bound method may be a useful in some settings as a simple approach. The proposed tests can be used in different applications, including within-cluster resampling and combining exchangeable pp-values.