  Cr\'emer and McLean [1985] showed that, when buyers' valuations are drawn
from a correlated distribution, an auction with full knowledge on the
distribution can extract the full social surplus. We study whether this
phenomenon persists when the auctioneer has only incomplete knowledge of the
distribution, represented by a finite family of candidate distributions, and
has sample access to the real distribution. We show that the naive approach
which uses samples to distinguish candidate distributions may fail, whereas an
extended version of the Cr\'emer-McLean auction simultaneously extracts full
social surplus under each candidate distribution. With an algebraic argument,
we give a tight bound on the number of samples needed by this auction, which is
the difference between the number of candidate distributions and the dimension
of the linear space they span.
