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Normal variance mixtures: Distribution, density and parameter estimation

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  • Hintz, Erik
  • Hofert, Marius
  • Lemieux, Christiane

Abstract

Efficient algorithms for computing the distribution function, (log-)density function and for estimating the parameters of multivariate normal variance mixtures are introduced. For the evaluation of the distribution function, randomized quasi-Monte Carlo (RQMC) methods are utilized in a way that improves upon existing methods proposed for the special case of normal and t distributions. For evaluating the log-density function, an adaptive RQMC algorithm that similarly exploits the superior convergence properties of RQMC methods is introduced. This allows the parameter estimation task to be accomplished via an expectation–maximization-like algorithm where all weights and log-densities are numerically estimated. Numerical examples demonstrate that the suggested algorithms are quite fast. Even for high dimensions around 1000 the distribution function can be estimated with moderate accuracy using only a few seconds of run time. Also, even log-densities around −100 can be estimated accurately and quickly. An implementation of all algorithms presented is available in the R package nvmix (version ≥0.0.4).

Suggested Citation

  • Hintz, Erik & Hofert, Marius & Lemieux, Christiane, 2021. "Normal variance mixtures: Distribution, density and parameter estimation," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    • Handle: RePEc:eee:csdana:v:157:y:2021:i:c:s0167947321000098
    DOI: 10.1016/j.csda.2021.107175
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    References listed on IDEAS

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    1. 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.
    2. Shiyao Liu & Huaiqing Wu & William Q. Meeker, 2015. "Understanding and Addressing the Unbounded "Likelihood" Problem," The American Statistician, Taylor & Francis Journals, vol. 69(3), pages 191-200, August.
    3. Kotz,Samuel & Nadarajah,Saralees, 2004. "Multivariate T-Distributions and Their Applications," Cambridge Books, Cambridge University Press, number 9780521826549, October.
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