An optimal algorithm for Monte Carlo estimation

A typical approach to estimate an unknown quantity μ is to design an experiment that produces a random variable Z distributed in [O,1] with E[Z]=μ, run this experiment independently a number of times and use the average of the outcomes as the estimate. In this paper, we consider the case when no a priori information about Z is known except that is distributed in [0,1]. We describe an approximation algorithm AA which, given ε and δ, when running independent experiments with respect to any Z, produces an estimate that is within a factor 1+ε of μ with probability at least 1-δ. We prove that the expected number of experiments ran by AA (which depends on Z) is optimal to within a constant factor for every Z