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Moment Estimator for Random Vectors with Heavy Tails

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  • Meerschaert, Mark M.
  • Scheffler, Hans-Peter

Abstract

If a set of independent, identically distributed random vectors has heavy tails, so that the covariance matrix does not exist, there is no reason to expect that the sample covariance matrix conveys useful information. On the contrary, this paper shows that the eigenvalues and eigenvectors of the sample covariance matrix contain detailed information about the probability tails of the data. The eigen- vectors indicate a set of marginals which completely determine the moment behavior of the data, and the eigenvalues can be used to estimate the tail thickness of each marginal. The paper includes an example application to a data set from finance.

Suggested Citation

  • Meerschaert, Mark M. & Scheffler, Hans-Peter, 1999. "Moment Estimator for Random Vectors with Heavy Tails," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 145-159, October.
    • Handle: RePEc:eee:jmvana:v:71:y:1999:i:1:p:145-159
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    References listed on IDEAS

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    2. McCulloch, J Huston, 1997. "Measuring Tail Thickness to Estimate the Stable Index Alpha: A Critique," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(1), pages 74-81, January.
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    5. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Abry, Patrice & Didier, Gustavo, 2018. "Wavelet eigenvalue regression for n-variate operator fractional Brownian motion," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 75-104.
    2. Tsionas, Mike, 2012. "Simple techniques for likelihood analysis of univariate and multivariate stable distributions: with extensions to multivariate stochastic volatility and dynamic factor models," MPRA Paper 40966, University Library of Munich, Germany, revised 20 Aug 2012.
    3. Tsionas, Mike G., 2016. "Bayesian analysis of multivariate stable distributions using one-dimensional projections," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 185-193.
    4. Haijun Li, 2018. "Operator Tail Dependence of Copulas," Methodology and Computing in Applied Probability, Springer, vol. 20(3), pages 1013-1027, September.
    5. Becker-Kern, Peter & Pap, Gyula, 2008. "Parameter estimation of selfsimilarity exponents," Journal of Multivariate Analysis, Elsevier, vol. 99(1), pages 117-140, January.
    6. Tsionas, Efthymios G., 2012. "Estimating multivariate heavy tails and principal directions easily, with an application to international exchange rates," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1986-1989.
    7. Patrice Abry & B. Cooper Boniece & Gustavo Didier & Herwig Wendt, 2023. "Wavelet eigenvalue regression in high dimensions," Statistical Inference for Stochastic Processes, Springer, vol. 26(1), pages 1-32, April.

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