  It is well known that the conventional cumulative sum (CUSUM) test suffers
from low power and large detection delay. In order to improve the power of the
test, we propose two alternative statistics. The backward CUSUM detector
considers the recursive residuals in reverse chronological order, whereas the
stacked backward CUSUM detector sequentially cumulates a triangular array of
backwardly cumulated residuals. A multivariate invariance principle for partial
sums of recursive residuals is given, and the limiting distributions of the
test statistics are derived under local alternatives. In the retrospective
context, the local power of the tests is shown to be substantially higher than
that of the conventional CUSUM test if a break occurs in the middle or at the
end of the sample. When applied to monitoring schemes, the detection delay of
the stacked backward CUSUM is found to be much shorter than that of the
conventional monitoring CUSUM procedure. Furthermore, we propose an estimator
of the break date based on the backward CUSUM detector and show that in
monitoring exercises this estimator tends to outperform the usual maximum
likelihood estimator. Finally, an application of the methodology to COVID-19
data is presented.
