Let π1,...,πkbe k independent populations where we assume that the ith population distribution isThe parametersand σiϵ (0,∞) for i = l,...,k are all assumed unknown, but H1einf:,...,Hkare known positive definite p&z.Times;p matrices. We estimate parameters of the formwhere ci's are known nonzero constants for i = 1,...,k, by means of ellipsoidal confidence regions. Various two-stage and sequential procedures are proposed and some of their exact and asymptotic properties are studied. Statistical methods and some of their characteristics are discussed both when H1,...,Hkare simultaneously diagonalizable as well as when they are not.