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 c(n) that controls the type I error rate for the least favorable rho in [0,1]. The constant c(n) 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 c(n) = 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. Published by Elsevier Inc.