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Elicitation Complexity of Statistical Properties

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  • Rafael Frongillo
  • Ian A. Kash

Abstract

A property, or statistical functional, is said to be elicitable if it minimizes expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization. While recent work asks which properties are elicitable, we instead advocate for a more nuanced question: how many dimensions are required to indirectly elicit a given property? This number is called the elicitation complexity of the property. We lay the foundation for a general theory of elicitation complexity, including several basic results about how elicitation complexity behaves, and the complexity of standard properties of interest. Building on this foundation, our main result gives tight complexity bounds for the broad class of Bayes risks. We apply these results to several properties of interest, including variance, entropy, norms, and several classes of financial risk measures. We conclude with discussion and open directions.

Suggested Citation

  • Rafael Frongillo & Ian A. Kash, 2015. "Elicitation Complexity of Statistical Properties," Papers 1506.07212, arXiv.org, revised Aug 2020.
    • Handle: RePEc:arx:papers:1506.07212
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    References listed on IDEAS

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    4. Johanna F. Ziegel, 2013. "Coherence and elicitability," Papers 1303.1690, arXiv.org, revised Mar 2014.
    5. Wang, Ruodu & Ziegel, Johanna F., 2015. "Elicitable distortion risk measures: A concise proof," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 172-175.
    6. repec:hal:journl:hal-00921283 is not listed on IDEAS
    7. Gneiting, Tilmann & Raftery, Adrian E., 2007. "Strictly Proper Scoring Rules, Prediction, and Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 359-378, March.
    8. Hans Föllmer & Stefan Weber, 2015. "The Axiomatic Approach to Risk Measures for Capital Determination," Annual Review of Financial Economics, Annual Reviews, vol. 7(1), pages 301-337, December.
    9. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
    10. Tobias Fissler & Johanna F. Ziegel, 2015. "Higher order elicitability and Osband's principle," Papers 1503.08123, arXiv.org, revised Sep 2015.
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    Cited by:

    1. Rafael Frongillo, 2022. "Quantum Information Elicitation," Papers 2203.07469, arXiv.org.
    2. Tobias Fissler & Yannick Hoga, 2021. "Backtesting Systemic Risk Forecasts using Multi-Objective Elicitability," Papers 2104.10673, arXiv.org, revised Feb 2022.
    3. Christis Katsouris, 2021. "Optimal Portfolio Choice and Stock Centrality for Tail Risk Events," Papers 2112.12031, arXiv.org.
    4. Fissler Tobias & Ziegel Johanna F., 2021. "On the elicitability of range value at risk," Statistics & Risk Modeling, De Gruyter, vol. 38(1-2), pages 25-46, January.

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