Hypothesis Testing for the Parameters of a Covariance Matrix Having Linear Structure

We consider the general family of multivariate normal distributions where the mean vector lies in an arbitrary subspace and the covariance matrix E 0 has a linear structure. When the minimal sufficient statistic is complete, we derive test statistics having optimal properties for testing statistical hypothe-ses about 0. The important special case where the matrices in the covariance structure commute is emphasized. In this case X2-tests and F-tests are obtained. Several examples are worked out to indi-cate the wide range of applications covered by our investigations. In the process of constructing similar tests some results concerning the completeness of families of product measures are established. When one family is boundedly complete and the other is strongly complete the resulting product family is boundedly complete. Bounded completeness is strengthened to completeness provided that the strongly complete family is an exponential family