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statisticsmontecarlocorrectness

How do i prove that my derived equation and the Monte-Carlo simulation are equivalent?

I have derived and implemented an equation of an expected value. To show that my code is free of errors i have employed the Monte-Carlo computation a number of times to show that it converges into the same value as the equation that i derived.

As I have the data now, how can i visualize this? Is this even the correct test to do? Can I give a measure how sure i am that the results are correct?

Solution

  • Why not just do a simple t-test? From your theoretical equation, you have the true mean mu_0 and your simulators mean,mu_1. Note that we can't calculate mu_1, we can only estimate it using the mean/average. So our hypotheses are:

    H_0: mu_0 = mu_1  and H_1: mu_0 does not equal mu_1

    The test statistic is the usual one-sample test statistic, i.e.

    T = (mu_0 - x)/(s/sqrt(n))

    where

    • mu_0 is the value from your equation
    • x is the average from your simulator
    • s is the standard deviation
    • n is the number of values used to calculate the mean.

    In your case, n is going to be large, so this is equivalent to a Normal test. We reject H_0 when T is bigger/smaller than (-3, 3). This would be equivalent to a p-value < 0.01.

    A couple of comments:

    1. You can't "prove" that the means are equal.
    2. You mentioned that you want to test a number of values. One possible solution is to implement a Bonferroni type correction. Basically, you reduce your p-value to: p-value/N where N is the number of tests you are running.
    3. Make your sample size as large as possible. Since we don't have any idea about the variability in your Monte Carlo simulation it's impossible to say use n=....
    4. The value of p-value < 0.01 when T is bigger/smaller than (-3, 3) just comes from the Normal distribution.