Sampling algorithms for estimating the mean of bounded random variables (Q1861583)

A sequential procedure is proposed for estimation of the mean \(\mu\) of i.i.d. observations \(z_1\), \(z_2\),\dots with \(0\leq z_i\leq b\), such that the obtained estimator \(\bar\mu\) satisfies \(\Pr\{|\bar\mu-\mu|\leq\varepsilon\mu\}\geq 1-\delta\) for predefined accuracy \(\varepsilon\) and confidence level \(\delta\). It is shown that the expected number of observations needed to derive \(\varepsilon\) and \(\delta\) is \[ E[N]\leq(1+\varepsilon)b\ln(2/\delta_s)[(\mu((1+\varepsilon)\ln(1+\varepsilon)-\varepsilon)]^{-1}, \] where \(\delta_s\) is a quantity defined in the paper to be close to \(\delta\). A variant of the algorithm for the case where the variance of \(z_i\) is known is also considered. The estimation technique is based on new inequalities for the sample mean of bounded i.i.d. variables.