Systematic inaccuracy is inherent in any computational estimate of anon-linear average, such as the free energy difference (Delta-F) between twostates or systems, because of the availability of only a finite number of datavalues, N. In previous work, we outlined the fundamental statisticaldescription of this ``finite-sampling error.'' We now give a more completepresentation of (i) rigorous general bounds on the free energy and othernonlinear averages, which underscore the universality of the phenomenon; (ii)asymptotic N->infinity expansions of the average behavior of thefinite-sampling error in Delta-F estimates; (iii) illustrative examples oflarge-N behavior, both in free-energy and other calculations; and (iv) theuniversal, large-N relation between the average finite-sampling error and thefluctuation in the error. An explicit role is played by Levy and Gaussianlimiting distributions.
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