ABSTRACT
Let Xa=(Xa1,…, Xap)', a=1,…, N1, be a random sample of size N1 from a p-variate normal population with mean vector µ and positive definite covariance matrix S, and let Ya=(Ya1,…, Yap)
', a=1,…, N2, be a random sample of size N2 from another independent normal population with mean v and positive definite covariance matrix S. Let
It can be verified that is a complete sufficient statistic for (µ, v, S),
has p-variate normal distribution with mean (N1N2/
(N1+N2)) 1/2(µ-v) and positive definite covariance matrix S, and S is distributed as
Wishart Wp(N1+N2-2, S) independently of The problem of testing the hypothesis H0 : µ-v=0 against the alternatives H1: µ-v?0 remains invariant under
the group of affine transformations Xa? gXa+b, a=1,…, N1, Y a? gYa+b, a=1,…,
N2, where (Eudidean p-space). The maximal invariant under the
group of affine transformations in the space of is given by
and T2 is distributed as Hotelling’s T2 with N1+N2-2 degrees of freedom and the noncentrality parameter
An optimum test for this problem is the Hotelling’s T2-test which rejects H0 for
large values of T2. This test possesses all the properties of the T2-test discussed above.
