Confidence bounds for the mean in nonparametric multisample problems

In auditing practice it often occurs that a statement regarding the accounting error in a population consisting of several subpopulations has to be made. As the relative proportion of errors can differ dramatically across these subpopulations, it is desirable to take independent fixed-size dollar-unit samples from each of them, as this often leads to lower variability compared with dollar-unit sampling from the whole population. It also occurs that the results of the separate investigations of, e.g. different branches of one company need to be combined to make a statement on the bookkeeping quality in general. The problem of estimating the total accounting error is thus related to the problem of estimating linear combinations of the mean values corresponding to several families of identically distributed independent random variables. In this article, we propose several confidence upper bounds for such linear combinations based on Hoeffding-type inequalities and show how they can be applied to the actual auditing problems. Simulation results comparing these modifications to the Hoeffding-based bounds for the one-sample case are also provided. It must be emphasized that the technique that we propose in this paper is fully justified from a mathematical point of view. Although the simulations show the proposed bounds to be highly conservative, they still present great interest, since we are not aware of any other method for estimation of the total accounting error in the multisample setting. Moreover, it is shown that significant improvements are hardly possible given the present conditions.