Gaussian copula of stable random vectors and application

Abstract

In this paper, we present a new method to investigate data of multivariate heavy-tailed distributions. We show that for any given number $\alpha \in (0;2]$, each Gaussian copula is also the copula of an $\alpha$-stable random vector. Simultaneously, every random vector is $\alpha$-stable if its marginals are $\alpha$-stable and its copula is a Gaussian copula. The result is used to build up a formula representing density functions of $\alpha$-stable random vectors with Gaussian copula. Adopting a new tool, the paper points out that pairs of GPS signals recording latitude and longitude of a fixed point have two-dimensional stable distribution, and in the most of cases, vectors of daily returns in stock market data have multivariate stable distributions with Gaussian copulas.

Keywords

• stable distributions,multivariate density function,GPSdata,stock market,portfolio selection

References

1. [1] R. J. Adler, R. E. Feldman, M. S Taqqu. A Practical Guide to Heavy Tailed Data, Birkhäuser, Boston, 1998.
2. [2] N. Bui Quang. On stable probability distributions and statistical application, Thesis, Academy of military science and technology, Ha Noi, 2016.
3. [3] P. Embrechts, F. Lindskog, A. McNeil. Modelling Dependence with Copulas and Applications to Risk Management, Handbook of Heavy Tailed Distributions in Finance, 2003, ed. Rachev S., Elsevier, Chapter 8, 329–384, 2001.
4. [4] E. Fama. The behavior of stock prices, Journal of Business 38, 34–105, 1965.
5. [5] C. Genest, B. Rémillard. Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models, Annales de lInstitut Henri Poincaré - Probabilités et Statistiques 44 no. 6, 1096–1127, 2008.
6. [6] S. M. Kogon, D. B. Williams. Characteristic function based estimation of stable parameters, In Adler R., Feldman R. and Taqqu M. (eds.) A Practical Guide to Heavy Tailed Data, Birkhäuser, Boston, MA, 311–335, 1998.
7. [7] I. Kojadinovic, J. Yan. Modeling Multivariate Distributions with Continuous Margins Using the copula R Package, J. Stat. Softw. 34 no. 9, 1–20, 2010.
8. [8] R. M. Kunst. Apparently stable increments in finance data: Could ARCH effects be the cause?, J. Statist. Comput. Simulation 45, 121–127, 1993.

9. [9] F. Lamantia, S. Ortobelli, S. Rachev. VaR, CVaR and Time Rules with Elliptical and Asymmetric Stable Distributed Returns, Investment Management and Financial Innovations 3, Issue 4, 19–39, 2006.
10. [10] J. H. McCulloch. Simple consistent estimators of stable distribution parameters, Comm. Statist. Simulation Comput. 15, 1109–1136, 1986.
11. [11] J. H. McCulloch. Financial applications of stable distributions, Handbook of Statistics 14, ed. G. Maddala and C. Rao, Elsevier Science Publishers, North-Holland, 393–425, 1996.
12. [12] J.P. Nolan. Multivariate elliptically contoured stable distributions: theory and estimation, Comput. Statist. 28, Issue 5, 2067–2089, 2013.
13. [13] J.P. Nolan. Stable Distributions - Models for Heavy Tailed Data, Birkhauser: Boston, MA, USA, 2016.
14. [14] K. J. Palmer, M. S. Ridout, B. J. T. Morgan. Modelling cell generation times using the tempered stable distribution, J. R. Stat. Soc. Ser. C. Appl. Stat. 57, 379–397, 2008.
15. [15] G. Samorodnitsky, M. S. Taqqu. Stable Non-Gaussian Random Processes, New York, NY, Chapman & Hall, 1994.
16. [16] P. Samuelson. Efficient portfolio selection for Pareto - Lévy investments, J. Financ. Quant. Anal. 2, 107–117, 1967.
17. [17] A. Sklar. Fonctions de rèpartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Universitè de Paris 8, 229–231, 1959.
18. [18] M. S. Taqqu. The modeling of Ethernet data and of signals that are heavy-tailed with infinite variance. Scand. J. Stat. 829, 273–295, 2002.