Rigorous error control methods for estimating means of bounded random variables

•A rigorous sample size method for estimating the means of bounded random variables.•It requires neither information nor IID condition of the underlying distribution.•It involves no approximation.•Sample complexity can be significantly reduced by using a mixed error criterion.•Explicit sample size formulae to ensure the statistical accuracy of estimation.

In this article, we propose rigorous sample size methods for estimating the means of random variables, which require no information of the underlying distributions except that the random variables are known to be bounded in a certain interval. Our sample size methods can be applied without assuming that the samples are identical and independent. Moreover, our sample size methods involve no approximation. We demonstrate that the sample complexity can be significantly reduced by using a mixed error criterion. We derive explicit sample size formulae to ensure the statistical accuracy of estimation.