Full surplus extraction from samples

We study whether an auctioneer who has only partial knowledge of the distribution of buyers' valuations can extract the full surplus. There is a finite number of possible distributions, and the auctioneer has access to a finite number of samples (independent draws) from the true distribution. Full surplus extraction is possible if the number of samples is at least the difference between the number of distributions and the dimension of the linear space they span, plus one. This bound is tight. The mechanism that extracts the full surplus uses the samples to construct contingent payments, and not for statistical inference.