For symmetric auctions, there is a close relationship between distributions of or-
der statistics of bidders’ valuations and observable bids that is often used to esti-
mate or bound the valuation distribution, optimal reserve price, and other quanti-
ties of interest nonparametrically. However, we show that the functional mapping
from distributions of order statistics to their parent distribution is, in general, not
Lipschitz continuous and, therefore, introduces an irregularity into the estimation
problem. More specifically, we derive the optimal rate for nonparametric point es-
timation of, and bounds for, the private value distribution, which is typically sub-
stantially slower than the regular root-n rate. We propose trimming rules for the
nonparametric estimator that achieve that rate and derive the asymptotic distri-
bution for a regularized estimator. We then demonstrate that policy parameters
that depend on the valuation distribution, including optimal reserve price and
expected revenue, are irregularly identified when bidding data are incomplete.
We also give rates for nonparametric estimation of descending bid auctions and
strategic equivalents.
Keywords. Empirical auctions, order statistics, bounds, irregular identification,
uniform consistency.
JEL classification. C13, C14, D44.