ABSTRACT
The discussions presented in this and the following section are sketchy. For further study and details relevant references are given.
In testing statistical hypotheses, the principle of unbiasedness plays an important role in deriving a suitable test statistic in complex situations involving composite
hypotheses. A size a test is said to be unbiased for testing against
if for In many such problems the principle of unbiasedness and the principle of invariance seem to complement each other in the sense that each is successful in the cases in which the other is not. For example, it is well known that a uniformly most powerful unbiased test exists for testing the
hypothesis (specified) against the alternatives in a normal distribution with mean µ whereas the principle of invariance does not reduce the problem sufficiently far to ensure the existence of a uniformly most powerful invariant test. On the other hand, for problems involving general linear hypotheses there exists a uniformly most powerful invariant test (F-test) but no uniformly most powerful unbiased test exists if the null hypothesis has more than one degree of freedom. However, if both principles can be applied successfully, then they lead to the same (almost everywhere) optimum test. Consider the problem of testing
against the alternatives Let us assume that it is invariant under the group of transformations G. Let Ca be the class of unbiased tests
of size a(0<a<1). For any test define the test function by
Obviously if and only if Thus if the test is a unique (up to measure 0) uniformly most powerful unbiased test for this problem, then
Thus and have the same power function. Hence under the assumption of completeness of the sufficient statistic, is almost invariant. Therefore if there exists a uniformly most powerful almost invariant test we have
for Comparing this with the trivial level a invariant test we conclude that is also unbiased, and hence
for Thus from (3.17) and (3.18) it follows that and have the same power function. Since is unique almost everywhere.
