  We propose a location-adaptive self-normalization (SN) based test for change
points in time series. The SN technique has been extensively used in
change-point detection for its capability to avoid direct estimation of
nuisance parameters. However, we find that the power of the SN-based test is
susceptible to the location of the break and may suffer from a severe power
loss, especially when the change occurs at the early or late stage of the
sequence. This phenomenon is essentially caused by the unbalance of the data
used before and after the change point when one is building a test statistic
based on the cumulative sum (CUSUM) process. Hence, we consider leaving out the
samples far away from the potential locations of change points and propose an
optimal data selection scheme. Based on this scheme, a new SN-based test
statistic adaptive to the locations of breaks is established. The new test can
significantly improve the power of the existing SN-based tests while
maintaining a satisfactory size. It is a unified treatment that can be readily
extended to tests for general quantities of interest, such as the median and
the model parameters. The derived optimal subsample selection strategy is not
specific to the SN-based tests but is applicable to any method that relies on
the CUSUM process, which may provide new insights in the area for future
research.
