Analyzing skewed financial data using skew scale-shap mixtures of multivariate normal distributions

Document Type : Special Issue Dedicated to memory of Prof. Mahbanoo Tata

Authors

1 Department of Statistics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

2 Department of Statistics, Faculty of Basic Sciences, University of Hormozgan, Bandar Abbas, Iran.

Abstract

This paper introduces an innovative family of statistical models called the multivariate skew scale-shape mixtures of normal distributions. These models serve as a versatile tool in statistical analysis by efficiently characterizing the skewed and leptokurtic nature commonly observed in multivariate datasets. Their applicability shines in real-world scenarios where data often deviate from standard statistical assumptions due to the presence of outliers. We present an EM-type algorithm designed for maximizing likelihood estimation and evaluate the model's effectiveness through real-world data applications. Through rigorous testing against various datasets, we assess the performance and practicality of the proposed algorithm in real statistical scenarios. The results demonstrate the remarkable performance of this new family of distributions.

Keywords

Main Subjects

References

[1] Adcock, C., Eling, M., & Loper do, N. (2015) Skewed distributions in  -nance and actuarial science: a review. Europian Journal of Finance 21:1{29. https://doi.org/10.1080/1351847X.2012.720269
[2] Amiri, M., & Balakrishnan, N. (2022). Hessian and increasing-Hessian orderings of scale-shape mixtures of multivariate skew-normal distributions and applications. Journal of Computational and Applied Mathematics, 402, 113801.
https://doi.org/10.1016/j.cam.2021.113801
[3] Arellano-Valle, R. B., & Genton, M. G. (2005). On fundamental skew distributions. Journal of Multivariate Analysis, 96(1), 93-116. https://doi.org/10.1016/j.jmva.2004.10.002
[4] Azzalini, A., & Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press.
[5] Bodie, Z., & Kane, A. (2020). Essentials of investments.
[6] Branco, M.D., & Dey, D. k. (2001). A general class of multivariate skew-elliptical distributions. Journal of Multivariate Analysis, 79, 99{113. https://doi.org/10.1006/jmva.2000.1960
[7] Cont, R. (2007). Volatility clustering in  nancial markets: empirical facts and agentbased models. In Long memory in economics (pp. 289-309). Berlin, Heidelberg: Springer Berlin Heidelberg.
[8] Denkowska, A., & Wanat, S. (2020). A tail dependence-based MST and their topological indicators in modeling systemic risk in the European insurance sector. Risks, 8(2), 39. https://doi.org/10.3390/risks8020039
[9] Fabozzi, F. J., Markowitz, H. M., & Gupta, F. (2008). Portfolio selection. Handbook of  nance.
[10] Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of  nancial economics, 33(1), 3-56. https://doi.org/10.1016/0304-405X(93)90023-5
[11] Ferreira, C. S., Bolfarine, H., & Lachos, V. H. (2011). Skew scale mixtures of normal distributions: properties and estimation. Statistical Methodology, 8(2), 154-171. https://doi.org/10.1016/j.stamet.2010.09.001
[12] Ferreira, C. S., Lachos, V. H., & Bolfarine, H. (2016). Likelihood-based inference for multivariate skew scale mixtures of normal distributions. AStA Advances in Statistical Analysis, 100(4), 421-441. https://doi.org/10.1007/s10182-016-0266-z
[13] Gomez-Deniz, E., Caldern-Ojeda, E., & Gomez, H. W. (2022). Symmetric and Asymmetric Distributions: Theoretical Developments and Applications III. Symmetry, 14(10), 21-43. https://doi.org/10.3390/sym14102143
[14] Hardle, W. K., & Simar, L. (2019). Applied multivariate statistical analysis. Springer Nature.
[15] Harvey, C. R., & Siddique, A. (2000). Conditional skewness in asset pricing tests. The Journal of  nance, 55(3), 1263-1295. https://doi.org/10.1111/0022-1082.00247
[16] Jamalizadeh, A., & Lin, T. I. (2017). A general class of scale-shape mixtures of skewnormal distributions: properties and estimation. Computational Statistics, 32, 451-474. https://doi.org/10.1007/s00180-016-0691-1
[17] Jorion, P. (2007). Value at risk: the new benchmark for managing  nancial risk. McGraw-Hill.
[18] Junior, L. S., & Franca, I. D. P. (2012). Correlation of  nancial markets in times of crisis. Physica A: Statistical Mechanics and its Applications, 391(1-2), 187-208. https://doi.org/10.1016/j.physa.2011.07.023
[19] Kotz, S., Balakrishnan, N., & Johnson, N. L. (2004). Continuous multivariate distributions, Volume 1: Models and applications (Vol. 1). John Wiley & Sons.
[20] Liu, C.H., & Rubin, D.B. (1994). The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika, 81, 633{648. https://doi.org/10.1093/biomet/81.4.633
[21] Madan, D. B. (2020). Multivariate distributions for  nancial returns. International Journal of Theoretical and Applied Finance, 23(06), 2050041. https://doi.org/10.1142/S0219024920500417
[22] Mahdavi, A., Amirzadeh, V., Jamalizadeh, A., & Lin, T. I. (2021). A Multivariate exible skew-symmetric-normal distribution: Scale-shape mixtures and parameter estimation via selection representation. Symmetry, 13(8), 1343.
https://doi.org/10.3390/sym13081343
[23] Mardia, K. V., Kent, J. T., & Bibby, J. M. (1979). Multivariate Analysis. Academic Press.
[24] Mata, L. M., Nu~nez Mora, J. A., & Serrano Bautista, R. (2021). Multivariate Distribution in the Stock Markets of Brazil, Russia, India, and China. SAGE Open, 11(2), 21582440211009509. https://doi.org/10.1177/21582440211009509
[25] McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts, techniques and tools. Princeton university press.
[26] Meng, X. L.,& Rubin, D. B. (1993). Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika, 80, 267-278. https://doi.org/10.1093/biomet/80.2.267
[27] Mondal, S., Arellano-Valle, R. B., & Genton, M. G. (2023). A multivariate modi ed skew-normal distribution. Statistical Papers, 1-45. https://doi.org/10.1007/s00362-023-01397-1
[28] Naderi, M., Arabpour, A., & Jamalizadeh, A. (2018). Multivariate normal mean-variance mixture distribution based on Lindley distribution. Communications in Statistics-Simulation and Computation, 47, 1179{1192. https://doi.org/10.1080/03610918.2017.1307400
[29] Pourmousa, R., Jamalizadeh, A., & Rezapour, M. (2015). Multivariate normal mean{variance mixture distribution based on Birnbaum{Saunders distribution. Journal of Statistical Computation and Simulation, 85(13), 2736{2749. https://doi.org/10.1080/00949655.2014.937435
[30] Pu, T., Zhang, Y., & Yin, C. (2023). Generalized location-scale mixtures of elliptical distributions: De nitions and stochastic comparisons. Communications in Statistics-Theory and Methods, 1-25. https://doi.org/10.1080/03610926.2023.2165407
Journal of Mahani Mathematical Research

Articles in Press, Accepted Manuscript
Available Online from 15 June 2024

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History

  • Receive Date: 29 December 2023
  • Revise Date: 27 March 2024
  • Accept Date: 14 June 2024

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