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Reference priors for exponential families with simple quadratic variance function

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  • Consonni, Guido
  • Veronese, Piero
  • Gutiérrez-Peña, Eduardo

Abstract

Reference analysis is one of the most successful general methods to derive noninformative prior distributions. In practice, however, reference priors are often difficult to obtain. Recently developed theory for conditionally reducible natural exponential families identifies an attractive reparameterization which allows one, among other things, to construct an enriched conjugate prior. In this paper, under the assumption that the variance function is simple quadratic, the order-invariant group reference prior for the above parameter is found. Furthermore, group reference priors for the mean- and natural parameter of the families are obtained. A brief discussion of the frequentist coverage properties is also presented. The theory is illustrated for the multinomial and negative-multinomial family. Posterior computations are especially straightforward due to the fact that the resulting reference distributions belong to the corresponding enriched conjugate family. A substantive application of the theory relates to the construction of reference priors for the Bayesian analysis of two-way contingency tables with respect to two alternative parameterizations.

Suggested Citation

  • Consonni, Guido & Veronese, Piero & Gutiérrez-Peña, Eduardo, 2004. "Reference priors for exponential families with simple quadratic variance function," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 335-364, February.
  • Handle: RePEc:eee:jmvana:v:88:y:2004:i:2:p:335-364
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    References listed on IDEAS

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    1. E. Gutiérrez-Peña & A. Smith & José Bernardo & Guido Consonni & Piero Veronese & E. George & F. Girón & M. Martínez & G. Letac & Carl Morris, 1997. "Exponential and bayesian conjugate families: Review and extensions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 6(1), pages 1-90, June.
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    3. Guido Consonni & Piero Veronese, 2001. "Conditionally Reducible Natural Exponential Families and Enriched Conjugate Priors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(2), pages 377-406, June.
    4. Y. Yang, 1995. "Invariance of the reference prior under reparametrization," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 4(1), pages 83-94, June.
    5. Manuel Mendoza, 1994. "Asymptotic normality under transformations. A result with Bayesian applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 3(2), pages 173-180, December.
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    Cited by:

    1. Ip, Edward H. & Wang, Yuchung J., 2008. "A note on cuts for contingency tables," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2356-2363, November.
    2. Piero Veronese & Eugenio Melilli, 2015. "Fiducial and Confidence Distributions for Real Exponential Families," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 471-484, June.
    3. Yacouba Boubacar Maïnassara & Célestin Kokonendji, 2014. "On normal stable Tweedie models and power-generalized variance functions of only one component," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 585-606, September.

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