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Tail dependence and smoothness of time series

Author

Listed:
  • Helena Ferreira

    (Universidade da Beira Interior (UBI))

  • Marta Ferreira

    (University of Minho
    University of Lisbon
    University of Lisbon)

Abstract

The risk of catastrophes is related to the possibility of occurring extreme values. Several statistical methodologies have been developed in order to evaluate the propensity of a process for the occurrence of high values and the permanence of these in time. The extremal index $$\theta $$ θ (Leadbetter in Z Wahrscheinlichkeitstheor Verw Geb 65:291–306, 1983) allows to infer the tendency for clustering of high values, but does not allow to evaluate the greater or less amount of oscillations in a cluster. The estimation of $$\theta $$ θ entails the validation of local dependence conditions regulating the distance between high levels oscillations of the process, which is difficult to implement in practice. In this work, we propose a smoothness coefficient to evaluate the degree of smoothness/oscillation in the trajectory of a process, with an intuitive reading and simple estimation. Application in some examples will be provided. We will see that, in a stationary sequence, it coincides with the tail dependence coefficient $$\lambda $$ λ (Sibuya in Ann Inst Stat Math 11:195–210, 1960; Joe in Multivariate models and dependence concepts. Monographs on statistics and applied probability, vol 73. Chapman and Hall, London, 1997), providing a new interpretation of the latter. This relationship will inspire a new estimator for $$\lambda $$ λ , and its performance will be evaluated based on a simulation study. We illustrate with an application to financial series.

Suggested Citation

  • Helena Ferreira & Marta Ferreira, 2021. "Tail dependence and smoothness of time series," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 198-210, March.
    • Handle: RePEc:spr:testjl:v:30:y:2021:i:1:d:10.1007_s11749-020-00709-z
    DOI: 10.1007/s11749-020-00709-z
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    References listed on IDEAS

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    1. Rafael Schmidt & Ulrich Stadtmüller, 2006. "Non‐parametric Estimation of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 307-335, June.
    2. Einmahl, J.H.J. & Krajina, A. & Segers, J., 2011. "An M-Estimator for Tail Dependence in Arbitrary Dimensions," Discussion Paper 2011-013, Tilburg University, Center for Economic Research.
    3. Ferreira, Helena & Ferreira, Marta, 2018. "Multidimensional extremal dependence coefficients," Statistics & Probability Letters, Elsevier, vol. 133(C), pages 1-8.
    4. Frahm, Gabriel & Junker, Markus & Schmidt, Rafael, 2005. "Estimating the tail-dependence coefficient: Properties and pitfalls," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 80-100, August.
    5. M. Ivette Gomes & Armelle Guillou, 2015. "Extreme Value Theory and Statistics of Univariate Extremes: A Review," International Statistical Review, International Statistical Institute, vol. 83(2), pages 263-292, August.
    6. Ferreira, Helena & Ferreira, Marta, 2014. "Extremal behavior of pMAX processes," Statistics & Probability Letters, Elsevier, vol. 93(C), pages 46-57.
    7. Marta Ferreira & Helena Ferreira, 2012. "On extremal dependence: some contributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(3), pages 566-583, September.
    8. Li, Haijun, 2009. "Orthant tail dependence of multivariate extreme value distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 243-256, January.
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