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Accurate bias estimation with applications to focused model selection

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  • Ingrid Dæhlen
  • Nils Lid Hjort
  • Ingrid Hobæk Haff

Abstract

We derive approximations to the bias and squared bias with errors of order o(1/n) where n is the sample size. Our results hold for a large class of estimators, including quantiles, transformations of unbiased estimators, maximum likelihood estimators in (possibly) incorrectly specified models, and functions thereof. Furthermore, we use the approximations to derive estimators of the mean squared error (MSE) which are correct to order o(1/n). Since the variance of many estimators is of order O(1/n), this level of precision is needed for the MSE estimator to properly take the variance into account. We also formulate a new focused information criterion (FIC) for model selection based on the estimators of the squared bias. Lastly, we illustrate the methods on data containing the number of battle deaths in all major inter‐state wars between 1823 and the present day. The application illustrates the potentially large impact of using a less‐accurate estimator of the squared bias.

Suggested Citation

  • Ingrid Dæhlen & Nils Lid Hjort & Ingrid Hobæk Haff, 2024. "Accurate bias estimation with applications to focused model selection," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 51(2), pages 724-759, June.
    • Handle: RePEc:bla:scjsta:v:51:y:2024:i:2:p:724-759
    DOI: 10.1111/sjos.12696
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    References listed on IDEAS

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    1. Gerda Claeskens & Christophe Croux & Johan Van Kerckhoven, 2006. "Variable Selection for Logistic Regression Using a Prediction-Focused Information Criterion," Biometrics, The International Biometric Society, vol. 62(4), pages 972-979, December.
    2. Vinnie Ko & Nils Lid Hjort & Ingrid Hobæk Haff, 2019. "Focused information criteria for copulas," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 46(4), pages 1117-1140, December.
    3. Martin Jullum & Nils Lid Hjort, 2019. "What price semiparametric Cox regression?," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 25(3), pages 406-438, July.
    4. Claeskens,Gerda & Hjort,Nils Lid, 2008. "Model Selection and Model Averaging," Cambridge Books, Cambridge University Press, number 9780521852258, January.
    5. Céline Cunen & Nils Lid Hjort & Håvard Mokleiv Nygård, 2020. "Statistical sightings of better angels: Analysing the distribution of battle-deaths in interstate conflict over time," Journal of Peace Research, Peace Research Institute Oslo, vol. 57(2), pages 221-234, March.
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