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Edgeworth expansions for multivariate random sums

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  • Javed, Farrukh
  • Loperfido, Nicola
  • Mazur, Stepan

Abstract

The sum of a random number of independent and identically distributed random vectors has a distribution which is not analytically tractable, in the general case. The problem has been addressed by means of asymptotic approximations embedding the number of summands in a stochastically increasing sequence. Another approach relies on fitting flexible and tractable parametric, multivariate distributions, as for example finite mixtures. Both approaches are investigated within the framework of Edgeworth expansions. A general formula for the fourth-order cumulants of the random sum of independent and identically distributed random vectors is derived and it is shown that the above mentioned asymptotic approach does not necessarily lead to valid asymptotic normal approximations. The problem is addressed by means of Edgeworth expansions. Both theoretical and empirical results suggest that mixtures of two multivariate normal distributions with proportional covariance matrices satisfactorily fit data generated from random sums where the counting random variable and the random summands are Poisson and multivariate skew-normal, respectively.

Suggested Citation

  • Javed, Farrukh & Loperfido, Nicola & Mazur, Stepan, 2024. "Edgeworth expansions for multivariate random sums," Econometrics and Statistics, Elsevier, vol. 31(C), pages 66-80.
    • Handle: RePEc:eee:ecosta:v:31:y:2024:i:c:p:66-80
    DOI: 10.1016/j.ecosta.2021.04.005
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    1. Ronchetti, Elvezio, 2020. "Accurate and robust inference," Econometrics and Statistics, Elsevier, vol. 14(C), pages 74-88.
    2. Marco Corazza & Florence Legros & Cira Perna & Marilena Sibillo, 2017. "Mathematical and Statistical Methods for Actuarial Sciences and Finance," Post-Print hal-01776135, HAL.
    3. Bernardi, Mauro & Maruotti, Antonello & Petrella, Lea, 2012. "Skew mixture models for loss distributions: A Bayesian approach," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 617-623.
    4. Mutschler, Willi, 2018. "Higher-order statistics for DSGE models," Econometrics and Statistics, Elsevier, vol. 6(C), pages 44-56.
    5. Pavel Cizek & Wolfgang Karl Härdle & Rafal Weron, 2011. "Statistical Tools for Finance and Insurance (2nd edition)," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook1101, December.
    6. Arevalillo, Jorge M. & Navarro, Hilario, 2012. "A study of the effect of kurtosis on discriminant analysis under elliptical populations," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 53-63.
    7. Loperfido, Nicola, 2014. "A note on the fourth cumulant of a finite mixture distribution," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 386-394.
    8. Kollo, Tõnu, 2008. "Multivariate skewness and kurtosis measures with an application in ICA," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2328-2338, November.
    9. Bauwens, Luc & Braione, Manuela & Storti, Giuseppe, 2017. "A dynamic component model for forecasting high-dimensional realized covariance matrices," Econometrics and Statistics, Elsevier, vol. 1(C), pages 40-61.
    10. Eling, Martin, 2012. "Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models?," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 239-248.
    11. Morana, Claudio, 2019. "Regularized semiparametric estimation of high dimensional dynamic conditional covariance matrices," Econometrics and Statistics, Elsevier, vol. 12(C), pages 42-65.
    12. Yanagihara, Hirokazu, 2007. "A family of estimators for multivariate kurtosis in a nonnormal linear regression model," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 1-29, January.
    13. Nicola Loperfido, 2019. "Finite mixtures, projection pursuit and tensor rank: a triangulation," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(1), pages 145-173, March.
    14. Ambagaspitiya, Rohana S., 1999. "On the distributions of two classes of correlated aggregate claims," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 301-308, May.
    15. Yanagihara, Hirokazu & Tonda, Tetsuji & Matsumoto, Chieko, 2005. "The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 237-264, October.
    16. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
    17. Nicola Loperfido & Stepan Mazur & Krzysztof Podgórski, 2018. "Third cumulant for multivariate aggregate claim models," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2018(2), pages 109-128, February.
    18. Bolance, Catalina & Guillen, Montserrat & Pelican, Elena & Vernic, Raluca, 2008. "Skewed bivariate models and nonparametric estimation for the CTE risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 386-393, December.
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    Keywords

    Edgeworth expansion; Fourth cumulant; Random sum; Skew-normal;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General

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