brownian motion – Question on the choice of boundary in the CUSUM test when we make some resampling

Question on the choice of boundary in the CUSUM test when we make some resampling

We are considering to make a CUSUM test for some economical time series $X = (x_1,..,x_n)$. Suppose $X$ contains many noises which need to be removed in some ways. Simply speaking, the CUSUM test considers the empirical process,

$W_n = frac{1}{sigma sqrt{n}}sum_{i=1}^{n} (x_i – mu )$

and check if it hits the boundary corresponding to the Brownian bridge. In practice, for execution of the CUSUM test, we utilize an R package, strucchange. Now we have a question. The question is related to the scaling of boundary since we are considering to make a resampling or filtering raw data, and perform a CUSUM test, again.

Question is as follows: Do we need to make any rescaling the boundary when we make a filtering data? For instance, should we need to multiply the original boundary by $frac{1}{sqrt{m}}$ (m: sample length)? Since we are using the open source R package, strucchange in performing the CUSUM test, it is not so clear if we need to take care of such a rescaling factor of the boundary function.