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 n-vector of multivariate normal random variables with known variance and unknown common correlation ρ is zero. We derive the standardized likelihood ratio test for known ρ and explore different ways of proceeding with ρ unknown. We evaluate the performance of the standardized statistic where ρ is replaced with an estimate of ρ and determine the critical value c_n that controls the type I error rate for the least favorable ρ in [0,1]. The constant c_n increases with n and this procedure has pathological behavior if ρ depends on n and ρ_n converges to zero at a certain rate. As an alternate approach, we replace ρ with the upper limit of a (1−β_n) confidence interval chosen so that c_n=c for all n. We determine β_n so that the type I error rate is exactly controlled for all ρ in [0,1]. We also investigate a simpler approach where we bound the type I error rate. The former method performs well for all n 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 p-values.