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On Bayes factors for the linear model

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  • T S Shively
  • S G Walker

Abstract

SummaryWe show that the Bayes factor for testing whether a subset of coefficients are zero in the normal linear regression model gives the uniformly most powerful test amongst the class of invariant tests discussed in Lehmann & Romano (2005) if the prior distributions for the regression coefficients are in a specific class of distributions. The priors in this class can have any elliptical distribution, with a specific scale matrix, for the subset of coefficients that are being tested. We also show under mild conditions that the Bayes factor gives the uniformly most powerful invariant test only if the prior for the coefficients being tested is an elliptical distribution with this scale matrix. The implications are discussed.

Suggested Citation

  • T S Shively & S G Walker, 2018. "On Bayes factors for the linear model," Biometrika, Biometrika Trust, vol. 105(3), pages 739-744.
  • Handle: RePEc:oup:biomet:v:105:y:2018:i:3:p:739-744.
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    File URL: http://hdl.handle.net/10.1093/biomet/asy022
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    References listed on IDEAS

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    1. Andrew J. Womack & Luis León-Novelo & George Casella, 2014. "Inference From Intrinsic Bayes' Procedures Under Model Selection and Uncertainty," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1040-1053, September.
    2. F. Javier Girón & M. Lina Martínez & Elías Moreno & Francisco Torres, 2006. "Objective Testing Procedures in Linear Models: Calibration of the p‐values," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(4), pages 765-784, December.
    3. Liang, Feng & Paulo, Rui & Molina, German & Clyde, Merlise A. & Berger, Jim O., 2008. "Mixtures of g Priors for Bayesian Variable Selection," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 410-423, March.
    4. Valen E. Johnson & David Rossell, 2010. "On the use of non‐local prior densities in Bayesian hypothesis tests," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(2), pages 143-170, March.
    5. Casella, George & Moreno, Elias, 2006. "Objective Bayesian Variable Selection," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 157-167, March.
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