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On the Dangers of Modelling through Continuous Distributions : A Bayesian Perspective

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  • Fernández, C.
  • Steel, M.F.J.

    (Tilburg University, Center For Economic Research)

Abstract

We point out that Bayesian inference on the basis of a given sample is not always possible with continuous sampling models, even under a proper prior. The reason for this paradoxical situation is explained, and linked to the fact that any dataset consisting of point observations has zero probability under a continuous sampling distribution. A number of examples, both with proper and improper priors, highlight the issues involved. A solution is proposed through the use of set observations, which take into account the precision with which the data were recorded. Use of a Gibbs sampler makes the solution practically feasible. The case of independent sampling from (possibly skewed) scale mixtures of Normals is analysed in detail for a location-scale model with a commonly used noninformative prior. For Student-t sampling with unrestricted degrees of freedom the usual inference, based on point observations, is shown to be precluded whenever the sample contains repeated observations. We show that Bayesian inference based on set observations, however, is possible and illustrate this by an application to a skewed dataset of stock returns.
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Suggested Citation

  • Fernández, C. & Steel, M.F.J., 1997. "On the Dangers of Modelling through Continuous Distributions : A Bayesian Perspective," Discussion Paper 1997-05, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:53bef46d-6511-4d09-9018-d6192f9d29af
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    References listed on IDEAS

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    1. Eric Jacquier & Nicholas G. Polson & Peter Rossi, "undated". "Stochastic Volatility: Univariate and Multivariate Extensions," Rodney L. White Center for Financial Research Working Papers 19-95, Wharton School Rodney L. White Center for Financial Research.
    2. Fernández, C. & Steel, M.F.J., 1995. "Reference priors in non-normal location problems," Discussion Paper 1995-91, Tilburg University, Center for Economic Research.
    3. Fernández, C. & Steel, M.F.J., 1996. "On Bayesian Inference under Sampling from Scale Mixtures of Normals," Discussion Paper 1996-02, Tilburg University, Center for Economic Research.
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    8. Fernández, C. & Steel, M.F.J., 1996. "On Bayesian Modelling of Fat Tails and Skewness," Other publications TiSEM 0991c197-c9e8-4904-8119-3, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Cristiano Villa, 2017. "Bayesian estimation of the threshold of a generalised pareto distribution for heavy-tailed observations," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 95-118, March.
    2. Vallejos, Catalina A. & Steel, Mark F.J., 2017. "Incorporating unobserved heterogeneity in Weibull survival models: A Bayesian approach," Econometrics and Statistics, Elsevier, vol. 3(C), pages 73-88.

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