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 [rho] is zero. We derive the standardized likelihood ratio test for known [rho] and explore different ways of proceeding with [rho] unknown. We evaluate the performance of the standardized statistic where [rho] is replaced with an estimate of [rho] and determine the critical value cn that controls the type I error rate for the least favorable [rho] in [0,1]. The constant cn increases with n and this procedure has pathological behavior if [rho] depends on n and [rho]n converges to zero at a certain rate. As an alternate approach, we replace [rho] with the upper limit of a (1-[beta]n) confidence interval chosen so that cn=c for all n. We determine [beta]n so that the type I error rate is exactly controlled for all [rho] 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.