  We consider the problem of conducting a survey with the goal of obtaining an
unbiased estimator of some population statistic when individuals have unknown
costs (drawn from a known prior) for participating in the survey. Individuals
must be compensated for their participation and are strategic agents, and so
the payment scheme must incentivize truthful behavior. We derive optimal
truthful mechanisms for this problem for the two goals of minimizing the
variance of the estimator given a fixed budget, and minimizing the expected
cost of the survey given a fixed variance goal.
