On a family of test statistics for discretely observed diffusion processes

We consider parametric hypotheses testing for multidimensional ergodic diffusion processes observed at discrete time. We propose a family of test statistics related to the so called ϕϕ-divergence measures. It is proved that the test statistics in this family are all asymptotically distribution free and converge to the chi squared distribution. In the case of contiguous alternatives, it is also possible to study in detail the power function of the tests.Although all the tests in this family are asymptotically equivalent, the second order expansion of the test statistics suggest a different behavior for the power function of the test in finite samples. We show by the Monte Carlo analysis that, in the small sample case, the performance of the test depends on the choice of the function ϕϕ and on the statistical model. The simulations show further that there is no uniformly most powerful test in this class.