We consider a linear regression model and search sequentially for a change in the regression parameters or the error variance, respectively. We assume that the errors are centered, uncorrelated and satisfy a weak invariance principle. For both types of changes we use statistics based on cumulative sums (CUSUM) of the residuals and a new modified moving-sum-statistic (mMOSUM) which was introduced in Chen and Tian (2010). Their statistic uses a bandwith parameter $h \in (0,1)$ which multiplicatively scales the lower bound of the moving sum.
We correct an error in the paper by Chen and Tian (2010) showing that unlike their findings the true limit distribution depends on the parameter $h$. In the process, we also use a different weighting function for the open-end procedure as their choice is clearly linked to the error.
Moreover we consider statistics for the closed-end procedure where monitoring stops after a predefined time. We derive the limit distribution for this case under the null hypotheses for both types of changes and both CUSUM as well as mMOSUM statistics for very general boundary functions. Furthermore, we show that the methods have asymptotic power one under weak assumptions on the boundary functions and the changes.
Finally, the methods are illustrated and compared in a simulation study. For a change in mean, we additionally include the standard MOSUM-Statistic in the comparison, which has not yet been theoretically examined in the regression setup. Our simulations coincide with the result in Chen and Tian (2010) showing that a change point which occurs late in the monitoring period, is detected more often with the modified MOSUM-Statistic than the competing procedures, which usually also detects changes quicker than the CUSUM-Statistic. This is joint work with Claudia Kirch.