  The paper studies the problem of auction design in a setting where the
auctioneer accesses the knowledge of the valuation distribution only through
statistical samples. A new framework is established that combines the
statistical decision theory with mechanism design. Two optimality criteria,
maxmin, and equivariance, are studied along with their implications on the form
of auctions. The simplest form of the equivariant auction is the average bid
auction, which set individual reservation prices proportional to the average of
other bids and historical samples. This form of auction can be motivated by the
Gamma distribution, and it sheds new light on the estimation of the optimal
price, an irregular parameter. Theoretical results show that it is often
possible to use the regular parameter population mean to approximate the
optimal price. An adaptive average bid estimator is developed under this idea,
and it has the same asymptotic properties as the empirical Myerson estimator.
The new proposed estimator has a significantly better performance in terms of
value at risk and expected shortfall when the sample size is small.
