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An Expectile Strong Law of Large Numbers

Author

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  • Collin Philipps

    (Department of Economics and Geosciences, US Air Force Academy)

Abstract

We show that Kolmogorov's classical strong law of large numbers applies to all expectiles uniformly. The expectiles of a random sample converge almost surely (uniformly) to the true expectiles if and only if the true data generating process has a finite first moment. The result holds for expectile functions of scalar and vector-valued random variables and can be reformulated to state that the mean (or any expectile) of a random sample converges almost surely to the true mean (or expectile) if and only if any arbitrary expectile exists and is finite.

Suggested Citation

  • Collin Philipps, 2022. "An Expectile Strong Law of Large Numbers," Working Papers 2022-05, Department of Economics and Geosciences, US Air Force Academy.
    • Handle: RePEc:ats:wpaper:wp2022-5
    as

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    References listed on IDEAS

    as
    1. Johanna F. Ziegel, 2016. "Coherence And Elicitability," Mathematical Finance, Wiley Blackwell, vol. 26(4), pages 901-918, October.
    2. Jones, M. C., 1994. "Expectiles and M-quantiles are quantiles," Statistics & Probability Letters, Elsevier, vol. 20(2), pages 149-153, May.
    3. Abdelaati Daouia & Irène Gijbels & Gilles Stupfler, 2019. "Extremiles: A New Perspective on Asymmetric Least Squares," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(527), pages 1366-1381, July.
    4. Koenker, Roger, 1993. "When are Expectiles Percentiles?," Econometric Theory, Cambridge University Press, vol. 9(03), pages 526-527, June.
    5. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Expectile Regression; Quantile Regression; Strong Law of Large Numbers;
    All these keywords.

    JEL classification:

    • C0 - Mathematical and Quantitative Methods - - General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions

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