On Independence of Sample Mean and Translation Invariant Statistics of Samples from Multivariate Normal Populations.

The article presents information on independence of sample mean and translation invariant statistics of samples from multivariate normal populations. In this case it could be said that the sample has circularly symmetric type dependence. Statistical analysis of normal sample dependent in the preceding fashion has been considered in the multivariate case by I. Olkin and in the univariate case by L. Guttman. The main result is a theorem giving a necessary and sufficient condition for the independence of the sample mean and a translation invariant statistic of a normal sample. The sample mean is independent of any translation invariant statistic, if the sample is an equicorrelated normal sample or a normal sample with circularly symmetric type dependence.