ABSTRACT
Theorem 7.2.3. For each c=0 the rejection region X'(XX'+YY')- 1X=c or, equivalently, the T2-test is admissible for testing H0 : µ=0 against H1 : µ?0.
We shall now examine the minimax property of the T2-test for testing H0 : µ=0
against the alternatives Nµ'S-1µ>0. As shown earlier the full linear group G does not satisfy the conditions of the Hunt-Stein theorem. But the subgroup GT(GBT with k=p), the multiplicative group of p×p nonsingular lower triangular matrices which leaves the present problem invariant operating as
satisfies the conditions of the Hunt-Stein theorem (see Kiefer, 1957; or Lehmann, 1959, p. 345). We observed in Chapter 3 that on GT there exists a right invariant measure. Thus there is a test of level a which is almost invariant under GT, and hence in the present problem there is such a test which is invariant under GT and
which maximizes among all level a tests the minimum power over H1 Whereas T 2
was a maximal invariant under G with a single distribution under each of H0 and H1
for each d2, the maximal invariant under GT is the p-dimensional statistic
as defined in Section 7.2.1 with k=p, p1=···=pk=1, or its equivalent statistic (R1,…, Rp) as defined in Chapter 6 with k=p. The distribution of R=(R1,…, Rp) has been worked out in Chapter 6. As we have observed there, under
R has a single distribution, but under H1 with d 2 fixed, it
depends continuously on a (p-1)-dimensional vector
for each fixed d2. Thus for N>p>1 there is no uniformly most powerful invariant test under GT for testing H0 against H1: Nµ'
S-1µ>0. Let fR(r|? ), fR(r|0) denote the probability density function of R under H1
(for fixed d2) and H0, respectively. Because of the compactness of the reduced parametric spaces {0} under H0 and
under H1 and the continuity fR(r|? ) in ? , it follows that (see Wald, 1950) every minimax test for the reduced problem in terms of R is Bayes. In particular,
Hotelling’s test which rejects H0 whenever which has constant power on
each contour Nµ'S-1µ=d2 (fixed) and which is also GT invariant, maximizes the
minimum power over H1 for each fixed d 2 if and only if there is a probability
density measure ? on ? such that for some constant K
according as
except possibly for a set of measure 0. Obviously c depends on the level of significance a and the measure ? and the constant K may depend only on c and the specific value of d2. From (6.64) with k=p, we get
An examination of the integrand in this expression allows us to replace (7.43) by its equivalent
Clearly (7.43) implies (7.44). On the other hand, if there are a ? and a K for which
(7.44) is satisfied and if is such that writing f (r)=fR(r|? )/fR(r|0) and r**=cr*/c', we see at once that f(r*)= f(c'r**/c)>f(r**)=K,
because of the form of f and the fact that c'/c>1 and This and a similar argument for the case c'<c show that (7.44) implies (7.43). [Of course we do not
assert that the left-hand side of (7.44) still depends only on if ] The computations in the next section are somewhat simplified by the fact that for
fixed c and d2 we can at this point compute the unique value of K for which (7.44)
can possibly be satisfied. Let and write for the
version of the conditional Lebesgue density of given that which is
continuous in and u for ri>0, and is zero elsewhere. Write fU
(u|d2) for the probability density function of which depends on ? only through d2, and is continuous for 0<u<1 and vanishes elsewhere. Then (7.44) can be written as
for ri>0, The integral of (7.45), being a probability mixture of
probability densities, is itself a probability density in as is Hence the expression in brackets equals 1. It is well known that, for 0<c<1 (see
Theorem 6.8.1),
Hence (7.44) becomes
For p=2, N=3, writing
?* for the measure associated with ? on ? [?*(A)=?(d2A)] and noting that
we obtain from (7.46)
Writing
0=i<8 , for the ith moment of ?* we obtain from (7.47)
Giri, et al. (1963) after lengthy calculations showed that there exists an absolutely continuous probability measure ?* whose derivative m?(x) is given by
proving that for p=2, N=3, the T2-test is minimax for testing H0 against H1. Later Salaevskii (1968), using this reduction of the problem, after voluminous computations was able to show that there exists a probability measure ? for general p and N, establishing that the T2-test is minimax in general.
