ABSTRACT
It is well known that some simplification is introduced in a testing problem by characterizing the statistical tests as a function of the sufficient statistic and thus reducing the dimension of the sample space to the dimension of the space of the sufficient statistic. On the other hand, invariance by reducing the dimension of the sample space to that of the space of the maximal invariant also shrinks the parametric space. Thus a question naturally arises: Is it possible to use both principles simultaneously and if so in what order, i.e., first sufficiency and then invariance, or first invariance and then sufficiency. Under certain conditions this reduction can be done by using both principles, and the order in which the reduction is made is immaterial in such cases. The reader is referred to Hall et al. (1965) for these conditions and some related results.
