Suppose I have a function

f(x) = { 0 if 0 =< x < 0.5
{ Beta=0.5 if 0.5 =< x =< 1

y = f(x) + epsilon where epsilon ~AR(1)
i.e. epsilon_i = rho*epsilon_i-1 + Z_i
where Z_i ~ N(0,1) i.i.d


Now I'm trying to test the following null hypothesis.

H0: f(x) = c for each x in [0,1]
H1: f(x) is non-constant

So I decided to use the CUSUM statistic to detect the change point. To see that his is effective, I have to do a power study.

My concern is, How do I do the power study to make sure that the cusum statistic is effective? Can anyone give me tips/psuedocode/ outline as to how to do the power study on effectiveness on cusum.

I appreciate the help.