ABSTRACT
The condition (3.7) or its equivalent is known as the condition of invariance of probability distributions with respect to the transformation g on X.
Definition 3.2.1. Invariance of the parametric space O. The parametric space O remains invariant under a one-to-one transformation g: X onto X if the
induced transformation on O satisfies (i) for and (ii) for any there exists a such that
An equivalent way of writing (i) and (ii) is
If the P? for different values of ? are distinct, then is also one-to-one. Given a set of transformations, each leaving O invariant, the following theorem
will assert that we can always extend this set to a group G of transformations whose members also leave O invariant.
