This study presents a novel approach to the density estimation of private values from second-price auctions, diverging from the conventional use of smoothing-based estimators. We introduce a Grenander-type estimator, constructed based on a shape restriction in the form of a convexity constraint. This constraint corresponds to the renowned Myerson regularity condition in auction theory, which is equivalent to the concavity of the revenue function for selling the auction item. Our estimator is nonparametric and does not require any tuning parameters. Under mild assumptions, we establish the cube-root consistency and show that the estimator asymptotically follows the scaled Chernoff's distribution. Moreover, we demonstrate that the estimator achieves the minimax optimal convergence rate.

翻译：本研究提出了一种从第二价格拍卖中估计私有值密度的新方法，与传统的基于光滑化的估计方法不同。我们引入了一种基于凸性约束形状限制的格勒南德型估计量。该约束对应拍卖理论中著名的迈尔森正则条件，该条件等价于拍卖品收益函数的凹性。我们的估计量是非参数的，且无需任何调优参数。在温和假设下，我们证明了估计量的立方根一致性，并表明其渐近服从缩放后的切尔诺夫分布。此外，我们证明了该估计量达到了极小极大最优收敛速度。