We propose a new way to obtain identification results using order statistics as finite mixtures with two key properties: i) the weights are known integer numbers; and ii) the elements of the mixture are the distributions of the maximum over a subset of the original random variables. We leverage Exponentiated Distributions (ED), which extend extreme value theory results. ED are max-stable, and we show that finite mixtures of ED are linearly independent. This enables us to derive non-parametric identification results and extend commonly known results using Gumbel and Fréchet distributions, both examples of ED. The results have broad applications in auctions, discrete-choice, and other settings where maximum or minimum choices play a central role. We illustrate the usefulness of our results by proposing new estimators for auctions with bidder-level heterogeneity.
