Asymptotic power properties of the Cramér-von Mises test under contiguous alternatives

The asymptotic power of the Cramér-von Mises test (CvM test) when parameters are estimated from the data is studied under certain local (contiguous) alternatives. The notion of (asymptotic) direction and distance from the null hypothesis of alternatives is introduced, and it is shown that there exist directions with maximum, minimum, and arbitrary intermediate power. For each direction, there exists a natural asymptotic testing problem with an uniformly most powerful test that is compared with the CvM test. For that, the notion of asymptotic local efficiency (ALE) of the CvM test is introduced. finally, the influence of more information on the (unknown) parameter is studied for three tests of the CvM-type for independence. it is shown that for certain directions, a better knowledge of the parameter may increase the power, and for other ones decrease it. These properties are analogous to that of the X2-test in similar situations.