Estimating value in a uniform auction

Consider an auction in which increasing bids are made in sequence on an object whose value θ is known to each bidder. Suppose n bids are received, and the distribution of each bid is conditionally uniform. More specifically, suppose the first bid X1 is uniformly distributed on [0, θ], and the ith bid is uniformly distributed on [Xi−1, θ] for i = 2, …︁, n. A scenario in which this auction model is appropriate is described. We assume that the value θ is un known to the statistician and must be esimated from the sample X1, X2, …︁, Xn. The best linear unbiased estimate of θ is derived. The invariance of the estimation problem under scale transformations in noted, and the best invariant estimation problem under scale transformations is noted, and the best invariant estimate of θ under loss L(θ, a) = [(a/θ) − 1]2 is derived. It is shown that this best invariant estimate has uniformly smaller mean‐squared error than the best linear unbiased estimate, and the ratio of the mean‐squared errors is estimated from simulation experiments. A Bayesian formulation of the estimation problem is also considered, and a class of Bayes estimates is explicitly derived.