A Cramér–von Mises test for a class of mean time dependent CHARN models with application to change-point detection

We derive a Cramér–von Mises test for testing a class of time dependent coefficients Coditional Heteroscedastic AutoRegressive Non Linear (CHARN) models. The test statistic is based on the log-likelihood ratio process whose weak convergence in a suitable Fréchet space is studied under the null hypothesis and under the sequence of local alternatives considered. This study makes use of the locally asymptotically normal (LAN) result previously established. Using the Karhunen–Loève expansion of the limiting process of the log-likelihood ratio process, the asymptotic null distribution and the power of the test statistic are accurately approximated. These results are applied to change-point analysis. An empirical study is done for evaluating the performance of the methodology proposed.