I have observations of two random variables $S_1(t)$ and $S_2(t)$ and I'm trying to build a test to see if they are related. In other words I'm trying to test if unknown functions $F_1(t)$ and $F_2(t)$ are the same function. The parameters $r$ and $b$ are constants with $ r \geq 0$ and $n_1$ and $n_2$ are Gaussian noise.
$$ S_1(t) = F_1(t) + n_1 $$ $$ S_2(t) = rF_2(t) + b + n_2 $$
It's been a while since I've taken a statistics class and I'm having issues remembering how to build a test for this. I want to set $H_0: F_1(t) \neq F_2(t)$ and $H_1 : F_1(t) = F_2(t)$ but after that I'm getting lost.