For symmetric auctions, there is a close relationship between distributions of order statistics of bidders' valuations and observable bids that is often used to estimate or bound the valuation distribution, optimal reserve price, and other quantities 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 estimation of, and bounds for, the private value distribution, which is typically substantially slower than the regular root-n rate. We propose trimming rules for the nonparametric estimator that achieve that rate and derive the asymptotic distribution 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.The order statistics approach has been very fruitful for deriving nonparametric identification results and bounds for auction models. 1 However, as we show in this paper, the central step of "inverting out" the distribution of bidders' valuations from the distribution of an order statistic introduces an irregularity into the estimation problem. We show that point estimation of, or construction of bounds for, the cumulative distribution function (c.d.f.) of valuations from order statistics is generally at a rate slower than

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The objective of this article is to develop a parametric approach to estimating auctions with incomplete data using Extreme Value Theory (EVT). The methodology is mainly theoretical: we first review that, when only transaction prices can be observed, the distribution of private valuations is irregularly identified. The sample bias produced by nonparametric estimators will affect all functionals of practical interest. We provide simulations for a best-case scenario and a worst-case scenario. Our results show that, compared to nonparametric approaches, the approximation of such functionals developed using EVT produces more accurate results, is easy to compute, and does not require strong assumptions about the unobserved distribution of bidders' valuations. It is recommended that financial operators working with auctions use this parametric approach when facing incomplete datasets. Given the difficult nature of the analysis, this work does not provide large sample properties for the proposed estimators and recommends the use of bootstrapping. This article contributes originally to the literature of structural estimation of auction models providing a useful and robust parametric approximation.

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