Two tests for sequential detection of a change-point in a nonlinear model

• The real time change-point detection in a nonlinear model is studied.
• Two tests based on weighted CUSUM to detect a change-point are proposed.
• The nonlinearity modifies the results and the approach made in linear case.
• A generalization of the Hájek–Rényi inequality is established.
• By simulations, empirical powers are 1, for the two tests.

In this paper, two tests, based on weighted CUSUM of the least squares residuals, are studied to detect in real time a change-point in a nonlinear model. A first test statistic is proposed by extension of a method already used in the literature but for the linear models. It is tested under the null hypothesis, at each sequential observation, that there is no change in the model against a change presence. The asymptotic distribution of the test statistic under the null hypothesis is given and its convergence in probability to infinity is proved when a change occurs. These results will allow to build an asymptotic critical region. Next, in order to decrease the type I error probability, a bootstrapped critical value is proposed and a modified test is studied in a similar way. A generalization of the Hájek–Rényi inequality is established.Simulation results, using Monte-Carlo technique, for nonlinear models which have numerous applications, investigate the properties of the two statistic tests.