Local power of a Cramér–von Mises type test for parametric autoregressive models of order one

In this paper, we study the local power of a Cramér–von Mises type test for parametric autoregressive models, when the data are stationary and ergodic. Our test is based on the limiting distribution of the cumulative residual process associated to the null model. We prove the contiguity of the null hypothesis H0 and a sequence of local alternatives that converges to H0 at rate from a fixed direction. From this result, the limiting distribution of the test statistic and the power are computed under these local alternatives. Simulation experiments show that the test is powerful against some exponential models.