Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

On the second order property of elliptical multivariate regular variation

  • Moosup Kim (Department of Statistics, Keimyung University)
  • Received : 2024.01.14
  • Accepted : 2024.03.14
  • Published : 2024.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

Multivariate regular variation is a popular framework of multivariate extreme value analysis. However, a suitable parametric model needs to be introduced for efficient estimation of its spectral measure. In such a view, elliptical distributions have been employed for deriving such models. On the other hand, the second order behavior of multivariate regular variation has to be specified for investigating the property of the estimator. This paper derives such a behavior by imposing a widely adopted second order regular variation condition on the representation of elliptical distributions. As result, the second order variation for the convergence to spectral measure is characterized by a signed measure with a regular varying index. Moreover, it leads to the asymptotic bias of the estimator. For demonstration, multivariate t-distribution is considered.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT): Grant No. RS-2023-00243752.

References

  1. Bingham NH, Goldie CM, and Teugels JL (1987). Regular Variation, vol. 27 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge.
  2. Cai J-J, Einmahl JH, and De Haan L (2011). Estimation of extreme risk regions under multivariate regular variation, The Annals of Statistics, 39, 1803-1826.
  3. Einmahl JH, Yang F, and Zhou C (2020). Testing the multivariate regular variation model, Journal of Business & Economic Statistics, 39, 907-919.
  4. Hall P (1982). On some simple estimates of an exponent of regular variation, Journal of the Royal Statistical Society: Series B (Methodological), 44, 37-42.
  5. Hill BM (1975). A simple general approach to inference about the tail of a distribution, The Annals of Statistics, 3, 1163-1174.
  6. Hsing T (1991). On tail index estimation using dependent data, The Annals of Statistics, 19, 1547-1569.
  7. Hult H and Lindskog F (2002). Multivariate extremes, aggregation and dependence in elliptical distributions, Advances in Applied Probability, 34, 587-608.
  8. Joe H and Li H (2019). Tail densities of skew-elliptical distributions, Journal of Multivariate Analysis, 171, 421-435.
  9. Kim M (2021). Maximum likelihood estimation of spectral measure of heavy tailed elliptical distribution. In Proceedings of the Autumn Conference of the Korean Data & Information Science Society, Seoul, 119-119.
  10. Kim M (2022). Simulation of elliptical multivariate regular variation, The Korean Data & Information Science Society, 33, 347-357.
  11. Kluppelberg C, Kuhn G, and Peng L (2007). Estimating the tail dependence function of an elliptical distribution, Bernoulli, 13, 229-251.
  12. Li H and Hua L (2015). Higher order tail densities of copulas and hidden regular variation, Journal of Multivariate Analysis, 138 143-155.
  13. Resnick SI (2008). Extreme Values, Regular Variation, and Point Processes, Springer Science & Business Media, Berlin.
  14. Weller GB and Cooley D (2014). A sum characterization of hidden regular variation with likelihood inference via expectation-maximization, Biometrika, 101, 17-36.