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Prediction of Components in Random Sums

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  • Muneya Matsui

    (Nanzan University)

Abstract

We consider predictions of the random number and the magnitude of each iid component in a random sum based on its distributional structure, where only a total value of the sum is available and where iid random components are non-negative. The problem is motivated by prediction problems in a Poisson shot noise process. In the context, although conditional moments are best possible predictors under the mean square error, only a few special cases have been investigated because of numerical difficulties. We replace the prediction problem of the process with that of a random sum, which is more general, and establish effective numerical procedures. The methods are based on conditional technique together with the Panjer recursion and the Fourier transform. In view of numerical experiments, procedures work reasonably. An application in the compound mixed Poisson process is also suggested.

Suggested Citation

  • Muneya Matsui, 2017. "Prediction of Components in Random Sums," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 573-587, June.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:2:d:10.1007_s11009-016-9497-4
    DOI: 10.1007/s11009-016-9497-4
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    References listed on IDEAS

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    1. H. Panjer, Harry & Shaun Wang,, 1993. "On the Stability of Recursive Formulas," ASTIN Bulletin, Cambridge University Press, vol. 23(2), pages 227-258, November.
    2. Norberg, Ragnar, 1999. "Prediction of Outstanding Liabilities II. Model Variations and Extensions," ASTIN Bulletin, Cambridge University Press, vol. 29(1), pages 5-25, May.
    3. Matsui, Muneya, 2014. "Prediction in a non-homogeneous Poisson cluster model," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 10-17.
    4. Paul Embrechts & Marco Frei, 2009. "Panjer recursion versus FFT for compound distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 497-508, July.
    5. Jewell, William S., 1989. "Predicting Ibnyr Events and Delays: I. Continuous Time," ASTIN Bulletin, Cambridge University Press, vol. 19(1), pages 25-55, April.
    6. Yuanhui Xiao & Robert Lund, 2006. "Inference for Shot Noise," Statistical Inference for Stochastic Processes, Springer, vol. 9(1), pages 77-96, May.
    7. Norberg, Ragnar, 1993. "Prediction of Outstanding Liabilities in Non-Life Insurance1," ASTIN Bulletin, Cambridge University Press, vol. 23(1), pages 95-115, May.
    8. Panjer, Harry H., 1981. "Recursive Evaluation of a Family of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 22-26, June.
    9. Arjas, Elja, 1989. "The Claims Reserving Problem in Non-Life Insurance: Some Structural Ideas," ASTIN Bulletin, Cambridge University Press, vol. 19(2), pages 139-152, November.
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    Cited by:

    1. Mausumi Bose & Rahul Mukerjee, 2021. "Shorter prediction intervals for anonymous individual assessments in group decision-making via pairwise comparisons," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(3), pages 833-857, October.

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