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Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process

Author

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  • Pierpaolo De Blasi

    (University of Torino and Collegio Carlo Alberto)

  • Ramsés H. Mena

    (IIMAS Universidad Nacional Autónoma de México)

  • Igor Prünster

    (Bocconi University and BIDSA)

Abstract

Discrete random probability measures are a key ingredient of Bayesian nonparametric inference. A sample generates ties with positive probability and a fundamental object of both theoretical and applied interest is the corresponding number of distinct values. The growth rate can be determined from the rate of decay of the small frequencies implying that, when the decreasingly ordered frequencies admit a tractable form, the asymptotics of the number of distinct values can be conveniently assessed. We focus on the geometric stick-breaking process and we investigate the effect of the distribution for the success probability on the asymptotic behavior of the number of distinct values. A whole range of logarithmic behaviors are obtained by appropriately tuning the prior. A two-term expansion is also derived and illustrated in a comparison with a larger family of discrete random probability measures having an additional parameter given by the scale of the negative binomial distribution.

Suggested Citation

  • Pierpaolo De Blasi & Ramsés H. Mena & Igor Prünster, 2022. "Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(1), pages 143-165, February.
    • Handle: RePEc:spr:aistmt:v:74:y:2022:i:1:d:10.1007_s10463-021-00791-6
    DOI: 10.1007/s10463-021-00791-6
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    References listed on IDEAS

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    1. Hatjispyros, Spyridon J. & Merkatas, Christos & Nicoleris, Theodoros & Walker, Stephen G., 2018. "Dependent mixtures of geometric weights priors," Computational Statistics & Data Analysis, Elsevier, vol. 119(C), pages 1-18.
    2. De Blasi, Pierpaolo & Martínez, Asael Fabian & Mena, Ramsés H. & Prünster, Igor, 2020. "On the inferential implications of decreasing weight structures in mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 147(C).
    3. 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.
    4. François Caron & Emily B. Fox, 2017. "Sparse graphs using exchangeable random measures," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1295-1366, November.
    5. Gutiérrez, Luis & Gutiérrez-Peña, Eduardo & Mena, Ramsés H., 2014. "Bayesian nonparametric classification for spectroscopy data," Computational Statistics & Data Analysis, Elsevier, vol. 78(C), pages 56-68.
    6. Raffaele Argiento & Andrea Cremaschi & Marina Vannucci, 2020. "Hierarchical Normalized Completely Random Measures to Cluster Grouped Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(529), pages 318-333, January.
    7. 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.
    8. David B. Dahl & Ryan Day & Jerry W. Tsai, 2017. "Random Partition Distribution Indexed by Pairwise Information," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 721-732, April.
    9. Antonio Lijoi & Ramsés H. Mena & Igor Prünster, 2007. "A Bayesian Nonparametric Method for Prediction in EST Analysis," ICER Working Papers - Applied Mathematics Series 16-2007, ICER - International Centre for Economic Research.
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    Cited by:

    1. Hatjispyros, Spyridon J. & Merkatas, Christos & Walker, Stephen G., 2023. "Mixture models with decreasing weights," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).
    2. Iksanov, Alexander & Kotelnikova, Valeriya, 2022. "Small counts in nested Karlin’s occupancy scheme generated by discrete Weibull-like distributions," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 283-320.

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