Run Many Small or a Few Big Simulations to Estimate the Mean?

Question

I was just running some simulations on tossing a coin given certain conditions, to test out some ideas I had. I was trying to find the ratio $\frac{\mathtt{successful\ tosses}}{\mathtt{total\ tosses}}$. Coin tosses were iid.

At first, I was running simulations of 10,000 coin tosses, and I would have the program print the proportion of successful tosses.

Then I realized I could, at the expense of a few more seconds, run a simulation of 100,000 coin tosses. I knew that the standard error of the mean is inversely proportional to the square root of the sample size, so I thought that this might give me a more accurate result.

However, then I also thought: if I take ten simulations of 10,000 coin tosses each rather than 100,000 coin tosses, I will be able to estimate a normal distribution using those ten datasets, which would allow me to home in on the mean and thereby on the true ratio.

My question is: for the purpose of finding the aforementioned ratio, are these methods equivalent, or is there some fine difference between the two?

Answer 1

If you run 100,000 simulations, you'll get a very good estimate of the true (population) value of the parameter, but no information -- at least from the simulations -- on its sampling distribution (for instance, standard error). By running 10 times 10,000 simulations, you can also obtain information on the sampling distribution. (And, if you estimate the parameter by sample mean, you'll get to the same estimate as the mean of the 10 means will be the same as the mean of the pooled samples.)

You essentially discovered a core idea of resampling, more specifically, the jackknife. However, note that in practice it makes only sense if the sampling distribution is not known mathematically (in contrast to your example!).