Testing epidemic change in nearly nonstationary process with statistics based on residuals

On the observation of a sample of size n of a first order autoregressive process, we study the detection of an epidemic change in the mean of the innovations of this process. The autoregressive coefficient is either a constant in or may depend on n and tend, not too quickly, to 1 as n tends to infinity. Under the null hypothesis, the innovations are i.i.d. mean zero random variables, while under the alternative there is some unknown interval of time, whose length depends on n, during which their expectation is shifted by some common value . Since innovations are not observed, we build weighted scan statistics based on the least square residuals of the process. Assuming some tail conditions on the innovations, we find the limit distributions of the test statistics under no change and prove consistency for short change interval, e.g. whose length is of the order of for some .