Change point tests for gradual changes in the Poisson-INARCH(1) process

Change point tests are a common tool to identify structural changes in the distribution of time series. In recent years, there has been progress in detecting changes within times series in countable spaces, e.g. the natural numbers. Such time series can be modeled by using Poisson-INARCH processes. Generally, it is assumed that changes occur as abrupt changes between stationary distributions. But in practice such changes often occur as a smooth transition, called gradual changes. To include such a behavior, we propose a new intensity function for Poisson-INARCH processes which makes gradual changes possible. We moreover propose a gradual change by adding a deterministic time dependent term on the intensity function. Properties as stationarity and a strong mixing property are investigated for both approaches. Under the alternative, convergence of estimators is ensured. We use those properties to prove consistency of tests based on both approaches under the null hypothesis and the alternative. We then analyze the quality of the models based on a comparative simulation study.