Hypothesis testing if two random variables come from a related underlying function 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.  
 A: This is for if you flipped your hypothesis. (i.e. $H_{o}: F_{1}(t)=F_{2}(t)$) so maybe you can go off of this. 
You would use the Two Sample Kolmogorv_Smirnov Test. Which pretty much looks at the empirical cdfs of both samples and statistic is $D\dfrac{n_{1}+n_{2}}{n_{1}n_{2}}$ where $D$ is maximum distance between the two empirical cdfs (which will occur at one of the "jumps" of cdf) and $n_{i}$ is sample size for sample $i$. From this you would have to look at table of for distribution of this statistic and find pvalue from that. Here is more about it in detail including what distribution you would use to find pvalue for this test 
http://en.wikipedia.org/wiki/Kolmogorov–Smirnov_test#Two-sample_Kolmogorov.E2.80.93Smirnov_test
(Copy whole link and look at two sample)
A: I'm not sure this is a well-posed problem if the two functions are allowed to be arbitrary. I'm guessing from your notation that you actually know something about the form of $F_1$ and $F_2$ (for instance, why include $r$ and $b$, they could just be lumped in with $F_2$)?
So let's assume your functions $F$ take the same form, but are parameterized by parameters $\mu_1$ and $\mu_2$; in this case one way (not the only way, just one way) to test this hypothesis is by using a Lagrange multiplier test; specifically maximizing the joint likelihood of $S_1$ and $S_2$, subject to the constraint that $(\mu_1 - \mu_2)^2 > e$, for some suitably small $e$.
Obviously if the functions don't take the same form, this wouldn't work, strictly speaking. However, if they're not the same form, they also can't possibly be the same function...
