Determining Errors in Monte Carlo Simulation

I was wondering if anyone could throw light on possible errors associated with Monte Carlo sampling. I seem to be getting values that are slightly different each time despite running my model for 500,000 iterations and I was wondering how I could take account of this.

• 500k iterations can be a small number, depending on what you are simulating. Sampling tail events for example require huge sample sizes. What are you sampling? – Slug Pue Mar 26 '14 at 14:22
• @user126540 I am running a Monte Carlo Simulation where I sample from about 65 Normal Distributions (all with different means). I then put these values into an equation that gives me what I am looking for. I am basically doing this 500,000 times. I must also add that I am sampling uniformly from the Normal Distributions, but randomly select the samples that then go into the equation. – user131983 Mar 26 '14 at 14:28
• well, you will basically have 65 differently distributeed random variables (where each random variable is an average of 500k trials), so you have 65 Sources of randomness which will contribute to the variance of the outcome – Slug Pue Mar 26 '14 at 19:19
• @user126540 Is there some way of determining how they contribute to the variance in the outcome? – user131983 Mar 26 '14 at 19:23
• what sort of Equation are you putting the values into? – Slug Pue Mar 26 '14 at 19:34

1 Answer

If you are for example using a linear combination of the 65 distributions, one can look at the variance of the linear combination: If we let $\bar X_i$ be the average of $m$ simulations of the random variable $X_i$ (in this case $m$ is 500 000) and you have a linear combination $$c_1 \bar X_1 + c_2 \bar X_2 + \cdots +c_n \bar X_n$$ ($n$ is 65 in Your case) then $$Var(c_1 \bar X_1 + c_2 \bar X_2 + \cdots +c_n \bar X_n ) = \sum^n c_i^2Var(\bar X_i) = \sum^n c_i^2 \frac{\sigma^2_i}{m}$$ where $\sigma_i$ is the standard deviation of $X_i$. If for example all coefficients $c_i$ are $1$ and all variances are equal, this becomes $$\frac{n \sigma^2}{m}$$ so as you see, the variance increases linearly with the number of contributing variables.

• Thanks. May I ask what the c's refer to here? Also, what relationships between variables can be taken to be a Linear combination of the random variables? – user131983 Mar 26 '14 at 23:00
• @user131983 The $c_i$'s are just arbitrary coefficients. A linear combination is any sum in the form $$c_1 x_1 + c_2 x_2 + \cdots + c_n x_n$$ – Slug Pue Mar 27 '14 at 2:35