Study of asymptotically powerful parametric and non-parametric tests for bilinear autoregressive models.

Summary The Lagrange multiplier test appears to be a good tool to test diagonal bilinear models of order one. We use it to discriminate between linear autoregressive models of order one, and some subdiagonal bilinear models of order two, for which we give a necessary and sufficient condition of invertibility. We prove the contiguity of the null hypothesis and of a sequence of local alternatives, which allows us, thanks to the third lemma of le Cam, to obtain an explicit expression of the local theoretical power of the test. Numerical Monte Carlo simulations show that this power is well estimated by the experimental power. We also find that this test is good for testing the types of hypotheses considered. Parametric tests such as the Lagrange multiplier test, because they are built for specific parametric models, may lack robustness. Non-parametric tests to test the linearity of autoregressive models are few. In order to prepare extensions to more general autoregressive models, we construct on a compact of the set of reals, two non-parametric tests to test diagonal bilinear models of order one, stationary, geometrically alpha-mixing, and having noise with a fixed, unknown and bounded density law. The study of the asymptotic distribution of the test statistics, under the null hypothesis, is done using weak invariance principles. For each of these tests, using maximal inequalities, we exhibit a minorant of the power that converges to 1. We show that under local alternatives, the risk of the second kind error can be very close to one. When the noise is Gaussian, tests confirm these results, and prove at the same time that on the example of the diagonal bilinear model of order one, the Lagrange multiplier test is better than the two non-parametric tests.