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Grouped Normal Variance Mixtures

Author

Listed:
  • Erik Hintz

    (Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada)

  • Marius Hofert

    (Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada)

  • Christiane Lemieux

    (Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada)

Abstract

Grouped normal variance mixtures are a class of multivariate distributions that generalize classical normal variance mixtures such as the multivariate t distribution, by allowing different groups to have different (comonotone) mixing distributions. This allows one to better model risk factors where components within a group are of similar type, but where different groups have components of quite different type. This paper provides an encompassing body of algorithms to address the computational challenges when working with this class of distributions. In particular, the distribution function and copula are estimated efficiently using randomized quasi-Monte Carlo (RQMC) algorithms. We propose to estimate the log-density function, which is in general not available in closed form, using an adaptive RQMC scheme. This, in turn, gives rise to a likelihood-based fitting procedure to jointly estimate the parameters of a grouped normal mixture copula jointly. We also provide mathematical expressions and methods to compute Kendall’s tau, Spearman’s rho and the tail dependence coefficient λ . All algorithms presented are available in the R package nvmix (version ≥ 0.0.5).

Suggested Citation

  • Erik Hintz & Marius Hofert & Christiane Lemieux, 2020. "Grouped Normal Variance Mixtures," Risks, MDPI, vol. 8(4), pages 1-26, October.
    • Handle: RePEc:gam:jrisks:v:8:y:2020:i:4:p:103-:d:424439
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    References listed on IDEAS

    as
    1. Tu, Jun & Zhou, Guofu, 2011. "Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies," Journal of Financial Economics, Elsevier, vol. 99(1), pages 204-215, January.
    2. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
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