Résumé
We build a goodness-of-fit test of normality for the<br />innovations of an ARMA(p,q) model with known mean or trend. This <br />test is based on the data driven smooth test approach and is simple to<br />perform. An extensive simulation study is conducted to study the<br />behavior of the test for moderate sample sizes. Our<br />approach is generally more powerful than existing tests while holding<br />its level throughout most of the parameter space. This agrees with theoretical results showing the<br />superiority of the data driven smooth test approach in related<br />contexts.<br /><br />A semi-parametric test of independence (or serial<br />independence) is proposed between marginal vectors each of which is normally<br />distributed but without assuming the joint normality of these marginal<br />vectors. The test statistic is a Cramér-von Mises functional of a<br />process defined from the empirical characteristic function. This<br />process is defined similarly as the process of Ghoudi et al. (2001)<br />built from the empirical distribution function and used to test for<br />independence between univariate marginal variables. The test statistic<br />can be represented as a V statistic. It is consistent to detect any<br />form of dependence. The weak convergence of the process is<br />derived. The asymptotic distribution of the Cramér-von Mises<br />functionals is approximated by the Cornish-Fisher expansion using a<br />recursive formula for cumulants and by the numerical evaluations of<br />the eigenvalues in the inversion formula. The test statistic is<br />finally compared with Wilks' statistic for testing the parametric hypothesis of independence in the one-way MANOVA model with random effects.