  A problem of goodness-of-fit test for ergodic diffusion processes is
presented. In the null hypothesis the drift of the diffusion is supposed to be
in a parametric form with unknown shift parameter. Two Cramer-Von Mises type
test statistics are studied. The first one is based on local time estimator of
the invariant density, the second one is based on the empirical distribution
function. The unknown parameter is estimated via the maximum likelihood
estimator. It is shown that both the limit distributions of the two test
statistics do not depend on the unknown parameter, so the distributions of the
tests are asymptotically parameter free. Some considerations on the consistency
of the proposed tests and some simulation studies are also given.
