ABSTRACT
It may be remarked that the statistic (R1,…, Rk) defined in Chapter 6 is a one to
one transformation of and hence (R1,…, Rk) is also a maximal
invariant under GBT. The induced transformation on the parametric space O corresponding to GBT on X is identically equal to GBT and is defined by
Thus a corresponding maximal invariant in O under GBT is where
The problem of testing the hypothesis H0: µ=0 against the alternatives H1: µ?0 on
the basis of observations xa, a=1,…, N(N>p), remains invariant under the group G of linear transformations g (set of all p×p nonsingular matrices) which transform each xa to gxa. These transformations induce on the space of the sufficient statistic
the transformations
Obviously G=GBT if k=1 and p1=p. A maximal invariant in the space of is
The corresponding maximal invariant in the
parametric space O under G is (say). Its probability density function is given in (7.17). The following two theorems give the optimum character of the T2-test among the class of all invariant level a tests for H0 : µ=0. To state them we need the following definition of a statistical test.
