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Randomized approximation algorithms for many #P-complete problems (such asthe partition function of a Gibbs distribution, the volume of a convex body,the permanent of a $\{0,1\}$-matrix, and many others) reduce to creating randomvariables $X_1,X_2,\ldots$ with finite mean $\mu$ and standarddeviation$\sigma$ such that $\mu$ is the solution for the problem input, andthe relative standard deviation $|\sigma/\mu| \leq c$ for known $c$. Underthese circumstances, it is known that the number of samples from the $\{X_i\}$needed to form an $(\epsilon,\delta)$-approximation $\hat \mu$ that satisfies$\mathbb{P}(|\hat \mu - \mu| > \epsilon \mu) \leq \delta$ is at least$(2-o(1))\epsilon^{-2} c^2\ln(1/\delta)$. We present here an easy to implement$(\epsilon,\delta)$-approximation $\hat \mu$ that uses$(2+o(1))c^2\epsilon^{-2}\ln(1/\delta)$ samples. This achieves the same optimalrunning time as other estimators, but without the need for extra conditionssuch as bounds on third or fourth moments.

An optimal $(\epsilon,\delta)$-approximation scheme for the mean of random variables with bounded relative variance