I'm working on a problem that can be approached in two different ways. Both are Monte Carlo algorithms--but it's a hard problem, so I am unsure whether the expected values are indeed the same.
I need a way to figure out the probability that the expected values of both algorithms are the same, given the samples. I suspect that this will be fairly simple to find once the variance of both estimates is known.
So for this question, how can I get the variance of the Monte Carlo estimate without knowing the expected value? An estimate of the variance can be found by taking the sample variance, I think, but I'm not sure if that's helpful.
If you want to know the variance of the estimators under each method, then simply run each method, say, 1,000 times and record the resulting estimates. Now, calculate the variance of each of these sets of estimates and you're done.