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.



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