**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.