IDEAS home Printed from https://ideas.repec.org/a/spr/metrik/v84y2021i5d10.1007_s00184-020-00800-3.html
   My bibliography  Save this article

Convergence and inference for mixed Poisson random sums

Author

Listed:
  • Gabriela Oliveira

    (Universidade Federal de Minas Gerais)

  • Wagner Barreto-Souza

    (Universidade Federal de Minas Gerais
    King Abdullah University of Science and Technology)

  • Roger W. C. Silva

    (Universidade Federal de Minas Gerais)

Abstract

We study the limit distribution of partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixture between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between $$\alpha $$ α -stable distributions and NEF laws is established. We propose the estimation of the NEF model parameters through the method of moments and also by the maximum likelihood method via an Expectation–Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators, and an empirical illustration of the financial market is presented.

Suggested Citation

  • Gabriela Oliveira & Wagner Barreto-Souza & Roger W. C. Silva, 2021. "Convergence and inference for mixed Poisson random sums," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(5), pages 751-777, July.
    • Handle: RePEc:spr:metrik:v:84:y:2021:i:5:d:10.1007_s00184-020-00800-3
    DOI: 10.1007/s00184-020-00800-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00184-020-00800-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00184-020-00800-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Barreto-Souza, Wagner, 2015. "Long-term survival models with overdispersed number of competing causes," Computational Statistics & Data Analysis, Elsevier, vol. 91(C), pages 51-63.
    2. Christian Schluter & Mark Trede, 2016. "Weak convergence to the Student and Laplace distributions," Post-Print hal-01447853, HAL.
    3. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    4. Kozubowski, Tomasz J. & Rachev, Svetlozar T., 1994. "The theory of geometric stable distributions and its use in modeling financial data," European Journal of Operational Research, Elsevier, vol. 74(2), pages 310-324, April.
    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. Nuerxiati Abudurexiti & Kai He & Dongdong Hu & Svetlozar T. Rachev & Hasanjan Sayit & Ruoyu Sun, 2021. "Portfolio analysis with mean-CVaR and mean-CVaR-skewness criteria based on mean-variance mixture models," Papers 2111.04311, arXiv.org, revised Feb 2023.
    2. Lioudmila Vostrikova, 2020. "On Distributions Of Exponential Functionals Of The Processes With Independent Increments," Working Papers hal-01725776, HAL.
    3. Luca Pratelli & Pietro Rigo, 2021. "Convergence in Total Variation of Random Sums," Mathematics, MDPI, vol. 9(2), pages 1-11, January.
    4. Yiqing Chen & Jiajun Liu & Yang Yang, 2023. "Ruin under Light-Tailed or Moderately Heavy-Tailed Insurance Risks Interplayed with Financial Risks," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-26, March.
    5. Li, Jinzhu, 2016. "Uniform asymptotics for a multi-dimensional time-dependent risk model with multivariate regularly varying claims and stochastic return," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 195-204.
    6. He, Yue & Kawai, Reiichiro, 2022. "Moment and polynomial bounds for ruin-related quantities in risk theory," European Journal of Operational Research, Elsevier, vol. 302(3), pages 1255-1271.
    7. Christian Schluter & Mark Trede, 2019. "Size distributions reconsidered," Econometric Reviews, Taylor & Francis Journals, vol. 38(6), pages 695-710, July.
    8. Jing Wang & Zbigniew Palmowski & Corina Constantinescu, 2021. "How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability," Risks, MDPI, vol. 9(9), pages 1-17, August.
    9. Arash Fahim & Lingjiong Zhu, 2015. "Optimal Investment in a Dual Risk Model," Papers 1510.04924, arXiv.org, revised Feb 2023.
    10. Yuchao Dong & J'er^ome Spielmann, 2019. "Weak Limits of Random Coefficient Autoregressive Processes and their Application in Ruin Theory," Papers 1907.01828, arXiv.org, revised Feb 2020.
    11. Henrik Hult & Filip Lindskog, 2011. "Ruin probabilities under general investments and heavy-tailed claims," Finance and Stochastics, Springer, vol. 15(2), pages 243-265, June.
    12. Tomasz Kozubowski, 2000. "Exponential Mixture Representation of Geometric Stable Distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(2), pages 231-238, June.
    13. Harri Nyrhinen, 2015. "On real growth and run-off companies in insurance ruin theory," Papers 1511.01763, arXiv.org.
    14. Kozubowski, Tomasz J. & Panorska, Anna K., 1996. "On moments and tail behavior of v-stable random variables," Statistics & Probability Letters, Elsevier, vol. 29(4), pages 307-315, September.
    15. Fu, Ke-Ang & Ng, Cheuk Yin Andrew, 2017. "Uniform asymptotics for the ruin probabilities of a two-dimensional renewal risk model with dependent claims and risky investments," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 227-235.
    16. Dong, Y. & Spielmann, J., 2020. "Weak limits of random coefficient autoregressive processes and their application in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 1-11.
    17. Massing, Till & Puente-Ajovín, Miguel & Ramos, Arturo, 2020. "On the parametric description of log-growth rates of cities’ sizes of four European countries and the USA," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 551(C).
    18. Fleten, Stein-Erik & Lindset, Snorre, 2008. "Optimal hedging strategies for multi-period guarantees in the presence of transaction costs: A stochastic programming approach," European Journal of Operational Research, Elsevier, vol. 185(3), pages 1680-1689, March.
    19. Halvarsson, Daniel, 2013. "On the Estimation of Skewed Geometric Stable Distributions," Ratio Working Papers 216, The Ratio Institute.
    20. Yin, Chuancun & Wen, Yuzhen, 2013. "An extension of Paulsen–Gjessing’s risk model with stochastic return on investments," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 469-476.

    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:spr:metrik:v:84:y:2021:i:5:d:10.1007_s00184-020-00800-3. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.