Variance for Hit-and-Miss Monte Carlo is given by $Var(\theta)=\frac{\Theta*(1-\Theta)}{N}$ where $\theta$ is the estimated probability of Hit and N is the number of simulations. Can someone explain why? And what will be the variance when Importance Sampling is used?

  • $\begingroup$ How do you define importance sampling in connection with hit-and-miss Monte Carlo? With no restriction, the minimal variance of an importance sampling estimate is zero. $\endgroup$
    – Xi'an
    Apr 12 '13 at 20:14

The first is just the variance of $\hat{p} = X/n$ in a binomial.

If $X$ is the number of "hits" in $n$ trials,

$$\newcommand{\Var}{\operatorname{Var}}\Var(X) = n\theta(1-\theta)$$

(Wikipedia on the Binomial Distribution).


$$\Var(X/n) = \frac{1}{n^2} n\theta(1-\theta) = \frac{\theta(1-\theta)}{n}.$$

For a comparison of the variance between the plain binomial case and importance sampling, see this section of the Wikipedia article on importance sampling.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.