Combined Estimation of a Common Mean from Few Sample Sets and from Sample Sets of Rare Events

Given a number of independent realizations of the k-dimensional random variable x = (x₁, x₂,…, x_k), the components of which may be correlated or independent, each has the same marginal expectation. The question is how the componentwise averages over the realizations are combined to yield an unbiased nearly optimum estimate of the common mean, and how the variance of the mean is to be estimated. An answer is given for the extreme cases of a small number of realizations and of rare events, when the majority of realizations is meaningless and only a small fraction of the samples contributes effectively to the estimate. It is shown how the sample statistics, based on the maximum likelihood estimates, are corrected to yield unbiased estimates. The results can readily be applied in Monte Carlo calculations and in evaluations of experimental data.