In classical statistics and distribution testing, it is often assumed that elements can be sampled exactly from some distribution P, and that when an element x is sampled, the probability P(x) of sampling x is also known. In this setting, recent work in distribution testing has shown that many algorithms are robust in the sense that they still produce correct output if the elements are drawn from any distribution Q that is sufficiently close to P. This phenomenon raises interesting questions: under what conditions is a “noisy” distribution Q sufficient, and what is the algorithmic cost of coping with this noise? In this paper, we investigate these questions for the problem of estimating the sum of a multiset of N real values x1, . . ., xN. This problem is well-studied in the statistical literature in the case P = Q, where the Hansen-Hurwitz estimator [Annals of Mathematical Statistics, 1943] is frequently used. We assume that for some (known) distribution P, values are sampled from a distribution Q that is pointwise close to P. That is, there is a parameter γ < 1 such that for all xi, (1 − γ)P(i) ≤ Q(i) ≤ (1 + γ)P(i). For every positive integer k we define an estimator ζ_k for µ = ^P_i xi whose bias is proportional to γ^k (where our ζ1 reduces to the classical Hansen-Hurwitz estimator). As a special case, we show that if Q is pointwise γ-close to uniform and all xi ∈ {0, 1}, for any ε > 0, we can estimate µ to within additive error εN using m = Θ(N^{1− k1} /ε^2/k) samples, where k = ⌈(lg ε)/(lg γ)⌉. We then show that this sample complexity is essentially optimal. Interestingly, our upper and lower bounds show that the sample complexity need not vary uniformly with the desired error parameter ε: for some values of ε, perturbations in its value have no asymptotic effect on the sample complexity, while for other values, any decrease in its value results in an asymptotically larger sample complexity.