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Distributional Properties of means of Random Probability Measures

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  • Antonio Lijoi
  • Igor Pruenster

Abstract

The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.

Suggested Citation

  • Antonio Lijoi & Igor Pruenster, 2009. "Distributional Properties of means of Random Probability Measures," ICER Working Papers - Applied Mathematics Series 22-2009, ICER - International Centre for Economic Research.
    • Handle: RePEc:icr:wpmath:22-2009
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    File URL: http://www.bemservizi.unito.it/repec/icr/wp2009/ICERwp22-09.pdf
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    References listed on IDEAS

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    1. Lancelot F. James & Antonio Lijoi & Igor Prünster, 2009. "Posterior Analysis for Normalized Random Measures with Independent Increments," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 76-97, March.
    2. Guglielmi, Alessandra, 1998. "A simple procedure calculating the generalized Stieltjes transform of the mean of a Dirichlet process," Statistics & Probability Letters, Elsevier, vol. 38(4), pages 299-303, July.
    3. Torkel Erhardsson, 2008. "Non‐parametric Bayesian Inference for Integrals with respect to an Unknown Finite Measure," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(2), pages 369-384, June.
    4. Lijoi, Antonio & Mena, Ramses H. & Prunster, Igor, 2005. "Hierarchical Mixture Modeling With Normalized Inverse-Gaussian Priors," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1278-1291, December.
    5. Antonio Lijoi & Ramsés Mena & Igor Prünster, 2005. "Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior," Statistical Inference for Stochastic Processes, Springer, vol. 8(3), pages 283-309, December.
    6. Epifani, I. & Guglielmi, A. & Melilli, E., 2006. "A stochastic equation for the law of the random Dirichlet variance," Statistics & Probability Letters, Elsevier, vol. 76(5), pages 495-502, March.
    7. Nils Hjort & Andrea Ongaro, 2005. "Exact Inference for Random Dirichlet Means," Statistical Inference for Stochastic Processes, Springer, vol. 8(3), pages 227-254, December.
    8. Antonio Lijoi & Ramsés H. Mena & Igor Prünster, 2007. "Controlling the reinforcement in Bayesian non‐parametric mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(4), pages 715-740, September.
    9. Ilenia Epifani, 2003. "Exponential functionals and means of neutral-to-the-right priors," Biometrika, Biometrika Trust, vol. 90(4), pages 791-808, December.
    10. Lancelot F. James & Antonio Lijoi & Igor Prünster, 2006. "Conjugacy as a Distinctive Feature of the Dirichlet Process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(1), pages 105-120, March.
    11. Lancelot F. James & Antonio Lijoi & Igor Prünster, 2009. "On the posterior distribution of classes of random means," Carlo Alberto Notebooks 134, Collegio Carlo Alberto.
    12. Yano, Kouji & Yano, Yuko, 2008. "Remarks on the density of the law of the occupation time for Bessel bridges and stable excursions," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2175-2180, October.
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    Cited by:

    1. Antonio Lijoi & Igor Prünster, 2014. "Discussion of “On simulation and properties of the stable law” by L. Devroye and L. James," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 23(3), pages 371-377, August.
    2. Julyan Arbel & Riccardo Corradin & Bernardo Nipoti, 2021. "Dirichlet process mixtures under affine transformations of the data," Computational Statistics, Springer, vol. 36(1), pages 577-601, March.

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