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Interpreting statistical evidence by using imperfect models: robust adjusted likelihood functions

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  • Richard Royall
  • Tsung‐Shan Tsou

Abstract

Summary. The strength of statistical evidence is measured by the likelihood ratio. Two key performance properties of this measure are the probability of observing strong misleading evidence and the probability of observing weak evidence. For the likelihood function associated with a parametric statistical model, these probabilities have a simple large sample structure when the model is correct. Here we examine how that structure changes when the model fails. This leads to criteria for determining whether a given likelihood function is robust (continuing to perform satisfactorily when the model fails), and to a simple technique for adjusting both likelihoods and profile likelihoods to make them robust. We prove that the expected information in the robust adjusted likelihood cannot exceed the expected information in the likelihood function from a true model. We note that the robust adjusted likelihood is asymptotically fully efficient when the working model is correct, and we show that in some important examples this efficiency is retained even when the working model fails. In such cases the Bayes posterior probability distribution based on the adjusted likelihood is robust, remaining correct asymptotically even when the model for the observable random variable does not include the true distribution. Finally we note a link to standard frequentist methodology—in large samples the adjusted likelihood functions provide robust likelihood‐based confidence intervals.

Suggested Citation

  • Richard Royall & Tsung‐Shan Tsou, 2003. "Interpreting statistical evidence by using imperfect models: robust adjusted likelihood functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 391-404, May.
  • Handle: RePEc:bla:jorssb:v:65:y:2003:i:2:p:391-404
    DOI: 10.1111/1467-9868.00392
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    Cited by:

    1. Zhiwei Zhang, 2010. "Profile Likelihood and Incomplete Data," International Statistical Review, International Statistical Institute, vol. 78(1), pages 102-116, April.
    2. George Karabatsos, 2023. "Approximate Bayesian computation using asymptotically normal point estimates," Computational Statistics, Springer, vol. 38(2), pages 531-568, June.
    3. Tsung-Shan Tsou, 2011. "Likelihood inferences for the link function without knowing the true underlying distributions," Computational Statistics, Springer, vol. 26(3), pages 507-519, September.
    4. David T. Frazier & Christian P. Robert & Judith Rousseau, 2017. "Model Misspecification in ABC: Consequences and Diagnostics," Papers 1708.01974, arXiv.org, revised Jul 2019.
    5. Simon Vandekar & Ran Tao & Jeffrey Blume, 2020. "A Robust Effect Size Index," Psychometrika, Springer;The Psychometric Society, vol. 85(1), pages 232-246, March.
    6. Smith, Simon C. & Timmermann, Allan & Zhu, Yinchu, 2019. "Variable selection in panel models with breaks," Journal of Econometrics, Elsevier, vol. 212(1), pages 323-344.
    7. Hoch, Jeffrey S. & Blume, Jeffrey D., 2008. "Measuring and illustrating statistical evidence in a cost-effectiveness analysis," Journal of Health Economics, Elsevier, vol. 27(2), pages 476-495, March.
    8. Tsung-Shan Tsou, 2005. "Inferences of variance function - a parametric robust way," Journal of Applied Statistics, Taylor & Francis Journals, vol. 32(8), pages 785-796.
    9. John Copas & Shinto Eguchi, 2010. "Likelihood for statistically equivalent models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(2), pages 193-217, March.
    10. Shen, Chung-Wei & Tsou, Tsung-Shan & Balakrishnan, N., 2011. "Robust likelihood inference for regression parameters in partially linear models," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1696-1714, April.
    11. Li-Chu Chien & Tsung-Shan Tsou, 2014. "Robust influence diagnostics for generalized linear models with continuous responses," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 68(4), pages 324-343, November.
    12. P. G. Bissiri & C. C. Holmes & S. G. Walker, 2016. "A general framework for updating belief distributions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(5), pages 1103-1130, November.
    13. Tsung-Shan Tsou & Chi-Chuan Yang, 2015. "Universal surrogate likelihood functions for nonnegative continuous data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(6), pages 635-646, August.
    14. Caterina Conigliani & Andrea Tancredi, 2006. "Comparing parametric and semi-parametric approaches for bayesian cost-effectiveness analyses in health economics," Departmental Working Papers of Economics - University 'Roma Tre' 0064, Department of Economics - University Roma Tre.
    15. Li-Chu Chien, 2011. "A robust diagnostic plot for explanatory variables under model mis-specification," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(1), pages 113-126.
    16. Caterina Conigliani & Andrea Tancredi, 2009. "A Bayesian model averaging approach for cost‐effectiveness analyses," Health Economics, John Wiley & Sons, Ltd., vol. 18(7), pages 807-821, July.
    17. Lemonte, Artur J., 2013. "On the gradient statistic under model misspecification," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 390-398.

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