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Flat and Multimodal Likelihoods and Model Lack of Fit in Curved Exponential Families

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  • ROLF SUNDBERG

Abstract

. It is well known that curved exponential families can have multimodal likelihoods. We investigate the relationship between flat or multimodal likelihoods and model lack of fit, the latter measured by the score (Rao) test statistic WU of the curved model as embedded in the corresponding full model. When data yield a locally flat or convex likelihood (root of multiplicity >1, terrace point, saddle point, local minimum), we provide a formula for WU in such points, or a lower bound for it. The formula is related to the statistical curvature of the model, and it depends on the amount of Fisher information. We use three models as examples, including the Behrens–Fisher model, to see how a flat likelihood, etc. by itself can indicate a bad fit of the model. The results are related (dual) to classical results by Efron from 1978.

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  • Rolf Sundberg, 2010. "Flat and Multimodal Likelihoods and Model Lack of Fit in Curved Exponential Families," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 632-643, December.
    • Handle: RePEc:bla:scjsta:v:37:y:2010:i:4:p:632-643
    DOI: 10.1111/j.1467-9469.2010.00703.x
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    References listed on IDEAS

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    1. Drton, Mathias, 2008. "Multiple solutions to the likelihood equations in the Behrens-Fisher problem," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3288-3293, December.
    2. Mathias Drton, 2004. "Multimodality of the likelihood in the bivariate seemingly unrelated regressions model," Biometrika, Biometrika Trust, vol. 91(2), pages 383-392, June.
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    Cited by:

    1. Nanny Wermuth & Kayvan Sadeghi, 2012. "Sequences of regressions and their independences," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 215-252, June.

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