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Invariance, model matching and probability matching

Abstract

For an invariant statistical model we consider the induced right Haar prior distribution and the resulting right Haar posterior. Using this posterior distribution, HPD (highest posterior density) regions of constant posterior probability are constructed and shown to exhibit probability matching. Further, HPD regions for equivariant functions of the parameter are also shown to exhibit probability matching. A number of examples are considered.

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References

  • Berk, R.H. (1967). A special group structure and equivariant estimation. Ann. Math. Statist., 38, 1436–1445.

    MathSciNet  MATH  Article  Google Scholar 

  • Box, G.E.P and Tiao, G.C. (1973). Bayesian inference in statistical analysis. Addison-Wesley, Reading, MA.

    MATH  Google Scholar 

  • Eaton, M.L. (1983, 2007). Multivariate statistics: a vector space approach. Originally published by Wiley, New York. Reprinted by Institute of Mathematical Statistics as vol. 53 in LMNS, Beachwood, Ohio.

  • Eaton, M.L. (1989). Group invariance applications in statistics. In Regional Conference Series in Probability and Statistics, 1. Institute of Mathematical Statistics, Beachwood, Ohio.

    Google Scholar 

  • Eaton, M.L. (2008). Dutch book in simple multivariate normal prediction: another look. In Probability and Statistics: Essays in Honor of David A. Freedman. IMS Collections, 2 (D. Nolan and T. Speed, eds.). Institute of Matmematical Statistics, Beachwood, Ohio, pp 12–23.

    Chapter  Google Scholar 

  • Eaton, M.L. and Freedman, D.A. (2004). Dutch book against some “objective” priors. Bernoulli, 10, 861–872.

    MathSciNet  MATH  Article  Google Scholar 

  • Eaton, M.L., Muirhead, R.J. and Pickering, E.H. (2006). Assessing a vector of clinical observations. J. Statist. Plann. Inference, 136, 3383–3414.

    MathSciNet  MATH  Article  Google Scholar 

  • Eaton, M.L. and Sudderth, W.D. (1999). Consistency and strong inconsistency of group invariant predictive inferences. Bernoulli, 5, 833–854.

    MathSciNet  MATH  Article  Google Scholar 

  • Eaton, M.L. and Sudderth, W.D. (2001). Best invariant predictive inferences. In Algebraic Methods in Statistics and Probability (M. Viana and D. Richards, eds.). Contemp. Math., 287. American Mathematical Society, Providence, Rhode Island, pp 49–62.

  • Eaton, M.L. and Sudderth, W.D. (2002). Group invariant inference and right Haar measure. J. Statist. Plann. Inference, 103, 87–99.

    MathSciNet  MATH  Article  Google Scholar 

  • Eaton, M.L. and Sudderth, W.D. (2004). Properties of right Haar predictive inference. Sankhyā, 66, 487–512.

    MathSciNet  MATH  Google Scholar 

  • Eaton, M.L. and Sudderth, W.D. (2010). Invariance of posterior distributions under reparameterization. Sankhyā, 72-A, 101–118.

    MathSciNet  Article  Google Scholar 

  • Fisher, R.A. (1930). Inverse probability. Proc. Cambridge Philos. Soc., 26, 528–535.

    MATH  Article  Google Scholar 

  • Hooper, P. (1982). Invariant confidence sets with smallest expected measure. Ann. Statist., 10, 1283–1294.

    MathSciNet  MATH  Article  Google Scholar 

  • Hora, R.B. and Buehler, R.J. (1966). Fiducial theory and invariant estimation. Ann. Math. Statist., 37, 643–656.

    MathSciNet  MATH  Article  Google Scholar 

  • Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proc. R. Soc. Lond. Ser. A, 186, 453–461.

    MathSciNet  MATH  Article  Google Scholar 

  • Kiefer, J. (1957). Invariance, minimax sequential estimation, and continuous time processes. Ann. Math. Statist., 28, 573–601.

    MathSciNet  MATH  Article  Google Scholar 

  • Lehmann, E.L. (1986). Testing statistical hypotheses, second edition. Springer-Verlag, New York.

    MATH  Book  Google Scholar 

  • Lehmann, E.L. and Casella, G. (1998). Theory of point estimation, second edition. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Nachbin, L. (1965). The Haar integral. Van Nostrand, Princeton, N.J.

    MATH  Google Scholar 

  • Peisakoff, M. (1951). Transformation of Parameters. Unpublished Ph.D. Thesis. Princeton University.

  • Pitman, E.J.G. (1939). The estimation of the location and scale parameters of a continuous population of any given form. Biometrika, 30, 391–421.

    MATH  Google Scholar 

  • Robert, C.P. (1994). The Bayesian choice. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Severini, T.A., Mukerjee, R. and Ghosh, M. (2002). On an exact probability matching property of right-invariant priors. Biometrika, 89, 952–957.

    MathSciNet  MATH  Article  Google Scholar 

  • Stein, C. (1959). An examination of wide discrepancy between fiducial and confidence intervals. Ann. Math. Statist., 30, 877–880.

    MathSciNet  MATH  Article  Google Scholar 

  • Stein, C. (1965). Approximation of improper prior measures by prior probability measures. In Bernoulli (1713)-Bayes (1763)-LaPlace (1813), (L.M. LeCam and J. Neyman, eds.). Springer-Verlag, New York, pp 217–240.

    Chapter  Google Scholar 

  • Zhu, H. (2002). Invariant Predictive Inferences with Applications. Unpublished Ph.D. Thesis. University of Minnesota.

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Acknowledgements

Portions of M. L. Eaton’s research were supported by a grant from the University of Minnesota Retirees Association, Office of the Vice President for Research at the University of Minnesota.

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Correspondence to Morris L. Eaton.

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Eaton, M.L., Sudderth, W.D. Invariance, model matching and probability matching. Sankhya A 74, 170–193 (2012). https://doi.org/10.1007/s13171-012-0018-4

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  • DOI: https://doi.org/10.1007/s13171-012-0018-4

AMS (2000) subject classification

  • Primary 62F15
  • Secondary 62G15

Keywords and phrases

  • Invariance
  • credible regions
  • probability matching
  • confidence regions
  • equivariant functions