An unbiased likelihood ratio test for equality of the covariance matrices in several multivariate normal populations with partially known means

Let samples from d multivariate normal populations be given with unknown covariance matrices [Sigma]k, k = 1,..., and with the mean of the i'th sample (i = 1,...k)) in the k'th population given by Bkzk,i +ak,i where Bk is unknown and zk,i and ak,i are known. With unbiased estimates Qk of (nk - rk)[Sigma]k, for k = 1,..., d where rk = rank (zk,1... zk,nk), the test which rejects the hypothesis of equality of the covariance matrices for large values of the test statistic where n = [Sigma]1dnk and r = [Sigma]1drk, is unbiased against all alternatives.