Guerre, Emmanuel, Isabelle Perrigne, and Quang Vuong (2000), "Optimal Nonparametric Estimation of  First-Price Auctions." Econometrica, 68(3), 525-574.

This paper proposes a general approach and a computationally convenient estimation procedure for the structural analysis of auction data. Considering first-price sealed-bid auction models within the independent private value paradigm, we show that the underlying distribution of bidders' private values is identified from observed bids and the number of actual bidders without any parametric assumptions. Using the theoiy of minimax, we establish the best rate of uniform convergence at which the latent density of private values can be estimated nonparametrically from available data. We then propose a two-step kernel-based estimator that converges at the optimal rate.