  We define data transformations that leave certain classes of distributions
invariant, while acting in a specific manner upon the parameters of the said
distributions. It is shown that under such transformations the maximum
likelihood estimators behave in exactly the same way as the parameters being
estimated. As a consequence goodness--of--fit tests based on standardized data
obtained through the inverse of this invariant data--transformation reduce to
the case of testing a standard member of the family with fixed parameter
values. While presenting our results, we also provide a selective review of the
subject of equivariant estimators always in connection to invariant
goodness--of--fit tests. A small Monte Carlo study is presented for the special
case of testing for the Weibull distribution, along with real--data
illustrations.
