IDEAS home Printed from https://ideas.repec.org/p/icr/wpmath/29-2006.html

Distributions of Functionals of the two Parameter Poisson-Dirichlet Process

Author

Listed:
  • Lancelot F. James

  • Antonio Lijoi

  • Igor Prünster

Abstract

The present paper provides exact expressions for the probability distribution of linear functionals of the two–parameter Poisson–Dirichlet process. Distributional results that follow from the application of an inversion formula for a (generalized) Cauchy–Stieltjes transform are achieved. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean functional of a Poisson–Dirichlet process and the mean functional of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are illustrated. Our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics.

Suggested Citation

  • Lancelot F. James & Antonio Lijoi & Igor Prünster, 2006. "Distributions of Functionals of the two Parameter Poisson-Dirichlet Process," ICER Working Papers - Applied Mathematics Series 29-2006, ICER - International Centre for Economic Research.
    • Handle: RePEc:icr:wpmath:29-2006
    as

    Download full text from publisher

    File URL: http://www.bemservizi.unito.it/repec/icr/wp2006/ICERwp29-06.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ishwaran H. & James L. F, 2001. "Gibbs Sampling Methods for Stick Breaking Priors," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 161-173, March.
    2. Nils Hjort & Andrea Ongaro, 2005. "Exact Inference for Random Dirichlet Means," Statistical Inference for Stochastic Processes, Springer, vol. 8(3), pages 227-254, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Antonio Lijoi & Igor Prunster, 2009. "Models beyond the Dirichlet process," Quaderni di Dipartimento 103, University of Pavia, Department of Economics and Quantitative Methods.
    2. Stefano Favaro & Antonio Lijoi & Igor Prünster, 2012. "On the stick–breaking representation of normalized inverse Gaussian priors," DEM Working Papers Series 008, University of Pavia, Department of Economics and Management.
    3. Ryo Kato & Takahiro Hoshino, 2020. "Semiparametric Bayesian multiple imputation for regression models with missing mixed continuous–discrete covariates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(3), pages 803-825, June.
    4. Nils Lid Hjort & Andrea Ongaro, 2006. "On the distribution of random Dirichlet jumps," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(1), pages 61-92.
    5. Yuan Fang & Dimitris Karlis & Sanjeena Subedi, 2022. "Infinite Mixtures of Multivariate Normal-Inverse Gaussian Distributions for Clustering of Skewed Data," Journal of Classification, Springer;The Classification Society, vol. 39(3), pages 510-552, November.
    6. 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.
    7. Griffin, J. E. & Steel, M. F. J., 2004. "Semiparametric Bayesian inference for stochastic frontier models," Journal of Econometrics, Elsevier, vol. 123(1), pages 121-152, November.
    8. Inés M. Varas & Jorge González & Fernando A. Quintana, 2020. "A Bayesian Nonparametric Latent Approach for Score Distributions in Test Equating," Journal of Educational and Behavioral Statistics, , vol. 45(6), pages 639-666, December.
    9. Igari, Ryosuke & Hoshino, Takahiro, 2018. "A Bayesian data combination approach for repeated durations under unobserved missing indicators: Application to interpurchase-timing in marketing," Computational Statistics & Data Analysis, Elsevier, vol. 126(C), pages 150-166.
    10. Takahiro Hoshino & Ryosuke Igari, 2017. "Quasi-Bayesian Inference for Latent Variable Models with External Information: Application to generalized linear mixed models for biased data," Keio-IES Discussion Paper Series 2017-014, Institute for Economics Studies, Keio University.
    11. Liu Yang & Nandram Balgobin, 2022. "Sampling methods for the concentration parameter and discrete baseline of the Dirichlet Process," Statistics in Transition New Series, Statistics Poland, vol. 23(4), pages 21-36, December.
    12. Michael Braun & André Bonfrer, 2011. "Scalable Inference of Customer Similarities from Interactions Data Using Dirichlet Processes," Marketing Science, INFORMS, vol. 30(3), pages 513-531, 05-06.
    13. Annalina Sarra & Lara Fontanella & Simone Zio, 2019. "Identifying Students at Risk of Academic Failure Within the Educational Data Mining Framework," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 146(1), pages 41-60, November.
    14. Mark J. Jensen & John M. Maheu, 2018. "Risk, Return and Volatility Feedback: A Bayesian Nonparametric Analysis," JRFM, MDPI, vol. 11(3), pages 1-29, September.
    15. 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.
    16. Julyan Arbel & Stefano Favaro, 2021. "Approximating Predictive Probabilities of Gibbs-Type Priors," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 496-519, February.
    17. Bassetti, Federico & Casarin, Roberto & Leisen, Fabrizio, 2014. "Beta-product dependent Pitman–Yor processes for Bayesian inference," Journal of Econometrics, Elsevier, vol. 180(1), pages 49-72.
    18. Daniel R. Kowal & Antonio Canale, 2021. "Semiparametric Functional Factor Models with Bayesian Rank Selection," Papers 2108.02151, arXiv.org, revised May 2022.
    19. Mame Diarra Fall & Éric Barat, 2025. "A simple and efficient method for sampling mixture models based on Dirichlet and Pitman-Yor processes," Computational Statistics, Springer, vol. 40(8), pages 4675-4716, November.
    20. Guanyu Hu & Yishu Xue & Zhihua Ma, 2020. "Bayesian Clustered Coefficients Regression with Auxiliary Covariates Assistant Random Effects," Papers 2004.12022, arXiv.org, revised Aug 2021.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:icr:wpmath:29-2006. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Daniele Pennesi (email available below). General contact details of provider: https://edirc.repec.org/data/icerrit.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.