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Order Statistics as Finite Mixtures

Author

Listed:
  • JosŽ-Antonio Esp’n-S‡nchez

    (Yale University)

  • Charles Hodgson

    (Yale University)

  • Kevin OÕNeill

    (The MITRE Corporation)

Abstract

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 FrŽchet 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.

Suggested Citation

  • JosŽ-Antonio Esp’n-S‡nchez & Charles Hodgson & Kevin OÕNeill, 2025. "Order Statistics as Finite Mixtures," Cowles Foundation Discussion Papers 2455, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:2455
    as

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    File URL: https://cowles.yale.edu/sites/default/files/2025-08/d2455.pdf
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    References listed on IDEAS

    as
    1. Javier D. Donna & José†Antonio Espín†Sánchez, 2018. "Complements and substitutes in sequential auctions: the case of water auctions," RAND Journal of Economics, RAND Corporation, vol. 49(1), pages 87-127, March.
    2. Joachim Freyberger & Bradley J. Larsen, 2022. "Identification in ascending auctions, with an application to digital rights management," Quantitative Economics, Econometric Society, vol. 13(2), pages 505-543, May.
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