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

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  • Richard Royall
  • Tsung-Shan Tsou
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    Abstract

    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. Copyright 2003 Royal Statistical Society.

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    Bibliographic Info

    Article provided by Royal Statistical Society in its journal Journal of the Royal Statistical Society: Series B (Statistical Methodology).

    Volume (Year): 65 (2003)
    Issue (Month): 2 ()
    Pages: 391-404

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    Handle: RePEc:bla:jorssb:v:65:y:2003:i:2:p:391-404

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    Cited by:
    1. 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.
    2. 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.
    3. 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.
    4. Lemonte, Artur J., 2013. "On the gradient statistic under model misspecification," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 390-398.
    5. 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.
    6. 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.

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