On goodness-of-fit testing for ergodic diffusion process with shift parameter

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 test uses the local time estimator of the invariant density, the second one uses the empirical distribution function. The unknown parameter is estimated via the maximum likelihood estimator. It is shown that the limit distribution of the two test statistics does not depend on the unknown parameter, thus both the tests are asymptotically parameter free. Some considerations on the consistency of the proposed tests and some simulation studies are also given.