Application of Stable Random Vector with Gaussian Copula
Main Article Content
Abstract
More and more real-world datasets have heavy-tailed distribution, while the calculations for these distributions in multi-dimensional cases are complex. This work shows a method to investigate data of multivariate heavy-tailed distributions. The sufficient condition for every
a-stable random vector is that it has α-stable marginals and Gaussian copula. From that results, we have a procedure testing stable distribution of multi-dimensional data and a formula representing density functions of multivariate stable distribution. Adopted a new tool, datasets about daily returns of 4 stocks on HoSE and 3 grains were analyzed.
Keywords:
Multivariate stable distribution, Gaussian copula, daily returns data.
References
[1] R. M. Kunst, Apparently Stable Increments in Finance Data: Could ARCH Effects Be the Cause?, J. Statist. Comput. Simulation, Vol. 45, No. 1-2, 1993, pp. 121-127.
[2] 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, Vol. 57, No. 4, 2008, pp. 379-397.
[3] P. Samuelson, Efficient Portfolio Selection for Pareto - Lévy Investments, J. Financ. Quant. Anal, Vol. 2, No. 2, 1967, pp.107-117.
[4] M. S. Taqqu, The Modeling of Ethernet Data and of Signals That Are Heavy-tailed with Infinite Variance, Scand. J. Stat, Vol. 29, No. 2, 2002, pp. 273-295.
[5] P. Embrechts, F. Lindskog, A. McNeil, Modelling Dependence with Copulas and Applications to Risk Management, Handbook of Heavy Tailed Distributions in Finance, Elsevier, 2009, Chapter 8, pp. 329-384.
[6] J. P. Nolan, Multivariate Elliptically Contoured Stable Distributions: Theory and Estimation, Comput. Statist,
Vol. 28, No. 5, 2013, pp. 2067-2089.
[7] N. B. Quang, On Stable Probability Distributions and Statistical Application, Thesis, Academy of Military Science and Technology, Hanoi, 2016.
[8] J. H. McCulloch, Financial Applications of Stable Distributions, Handbook of Statistics 14, ed. G. Maddala and C. Rao, Elsevier Science Publishers, North-Holland, 1996, pp. 393-425.
[9] J. H. McCulloch, Simple Consistent Estimators of Stable Distribution Parameters, Comm. Statist. Simulation Comput, Vol. 15, No. 4, 1986, pp. 1109-1136.
[10] R. J. Adler, R. E. Feldman, M. S. Taqqu, A Practical Guide to Heavy Tailed Data, Birkhäuser, Boston, 1998.
[11] 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, 1998, pp. 311-335.
[12] J. P. Nolan, Stable Distributions - Models for Heavy Tailed Data, Birkhauser: Boston, MA, USA,
https://edspace.american.edu/jpnolan/wp-content/uploads/sites/1720/2020/09/Chap1.pdf/, 2018 (accessed on: December 15th, 2018).
[13] A. Sklar, Fonctions De Rèpartition à n Dimensions et Leurs Marges, Publications de l’Institut de Statistique de l’Universitè de Paris, Vol. 8, No. 3, 1959, pp. 229-231.
[14] F. Lamantia, S. Ortobelli, S. Rachev, VaR, CVaR and Time Rules with Elliptical and Asymmetric Stable Distributed Returns, Investment Management and Financial Innovations, Vol. 3, No. 4, 2006, pp. 19-39.
[15] H. D. Phuc, V. T. T. Giang, Gaussian Copula of Stable Random Vectors and Application, Hacet. J. Math. Stat,
Vol. 49, No. 2, 2020, pp. 887-901.
[16] J. Zolotarev, Empirical Characteristic Function Estimation and Its Applications, Econometric Reviews, Vol. 23, No. 2, 2004, pp. 93-123.
[17] G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes, New York, NY, Chapman & Hall, 1994.
[18] S. Corbella, D. D. Stretch, Simulating a Multivariate Sea Storm using Archimedean Copulas, Coast. Eng, Vol. 76, 2013, pp. 68-78.
[19] Q. Zhang, M. Xiao, V. P. Singh, X. Chen, Copula-based Risk Evaluation of Droughts Across the Pearl
River basin, China, Theoretical and Applied Climatology, Vol. 111, No. 1-2, 2013, pp. 119-131.
[20] R. P. Nelsen, An Introduction to Copulas, New York, Springer, 2006.
[2] 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, Vol. 57, No. 4, 2008, pp. 379-397.
[3] P. Samuelson, Efficient Portfolio Selection for Pareto - Lévy Investments, J. Financ. Quant. Anal, Vol. 2, No. 2, 1967, pp.107-117.
[4] M. S. Taqqu, The Modeling of Ethernet Data and of Signals That Are Heavy-tailed with Infinite Variance, Scand. J. Stat, Vol. 29, No. 2, 2002, pp. 273-295.
[5] P. Embrechts, F. Lindskog, A. McNeil, Modelling Dependence with Copulas and Applications to Risk Management, Handbook of Heavy Tailed Distributions in Finance, Elsevier, 2009, Chapter 8, pp. 329-384.
[6] J. P. Nolan, Multivariate Elliptically Contoured Stable Distributions: Theory and Estimation, Comput. Statist,
Vol. 28, No. 5, 2013, pp. 2067-2089.
[7] N. B. Quang, On Stable Probability Distributions and Statistical Application, Thesis, Academy of Military Science and Technology, Hanoi, 2016.
[8] J. H. McCulloch, Financial Applications of Stable Distributions, Handbook of Statistics 14, ed. G. Maddala and C. Rao, Elsevier Science Publishers, North-Holland, 1996, pp. 393-425.
[9] J. H. McCulloch, Simple Consistent Estimators of Stable Distribution Parameters, Comm. Statist. Simulation Comput, Vol. 15, No. 4, 1986, pp. 1109-1136.
[10] R. J. Adler, R. E. Feldman, M. S. Taqqu, A Practical Guide to Heavy Tailed Data, Birkhäuser, Boston, 1998.
[11] 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, 1998, pp. 311-335.
[12] J. P. Nolan, Stable Distributions - Models for Heavy Tailed Data, Birkhauser: Boston, MA, USA,
https://edspace.american.edu/jpnolan/wp-content/uploads/sites/1720/2020/09/Chap1.pdf/, 2018 (accessed on: December 15th, 2018).
[13] A. Sklar, Fonctions De Rèpartition à n Dimensions et Leurs Marges, Publications de l’Institut de Statistique de l’Universitè de Paris, Vol. 8, No. 3, 1959, pp. 229-231.
[14] F. Lamantia, S. Ortobelli, S. Rachev, VaR, CVaR and Time Rules with Elliptical and Asymmetric Stable Distributed Returns, Investment Management and Financial Innovations, Vol. 3, No. 4, 2006, pp. 19-39.
[15] H. D. Phuc, V. T. T. Giang, Gaussian Copula of Stable Random Vectors and Application, Hacet. J. Math. Stat,
Vol. 49, No. 2, 2020, pp. 887-901.
[16] J. Zolotarev, Empirical Characteristic Function Estimation and Its Applications, Econometric Reviews, Vol. 23, No. 2, 2004, pp. 93-123.
[17] G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes, New York, NY, Chapman & Hall, 1994.
[18] S. Corbella, D. D. Stretch, Simulating a Multivariate Sea Storm using Archimedean Copulas, Coast. Eng, Vol. 76, 2013, pp. 68-78.
[19] Q. Zhang, M. Xiao, V. P. Singh, X. Chen, Copula-based Risk Evaluation of Droughts Across the Pearl
River basin, China, Theoretical and Applied Climatology, Vol. 111, No. 1-2, 2013, pp. 119-131.
[20] R. P. Nelsen, An Introduction to Copulas, New York, Springer, 2006.