ABSTRACT
Let T2 be the transformation group which translates the last p3 components of each
Xa, and let GBT be as defined in Section 7.2.2 with k=3, p1+p2+p3=p. This problem remains invariant under the group (GBT, T2) of affine transformations, transforming
(with k=3), (t can be considered as a p-vector with the first p1+p2
components equal to zero). A maximal invariant in the space of [the induced
transformation on is is (R1, R2) [also its equivalent
statistic ]. A corresponding maximal invariant in O is Under H0,
and under the alternatives H1, From (7.81) it follows that the likelihood ratio test is not uniformly most
powerful (optimum) invariant for this problem and that there is no uniformly most powerful invariant test for the problem. However, for fixed p, the likelihood ratio test is nearly optimum as N becomes large (Wald, 1943). Thus, if p is not large, it seems likely that the sample size occurring in practice was usually large enough for this result to be relevant. However, if the dimension p is large, it may
be that the sample size N must be extremely large for this result to apply. Giri (1961) has shown that the difference of the powers of the likelihood ratio test and the best invariant test is o(N-1) when p1, p are both equal to O(N) and
For the minimax property Giri (1968) has shown that no invariant
test under (GBT, T2) is minimax for testing H0 against for every choice
of ?. However Giri (1968) has shown that the test which rejects against
the alternatives whenever R1+((n-p1)/ p2)R2=c where c depends on the level a of the test is locally best invariant and locally minimax as ?? 0.
