The generalized scale mixtures of asymmetric generalized normal distributions with application to stock data

Ruijie Guan; Aidi Liu; Weihu Cheng; Ruijie Guan; Aidi Liu; Weihu Cheng

doi:10.3934/math.2024064

AIMS Mathematics

2024, Volume 9, Issue 1: 1291-1322. doi: 10.3934/math.2024064

Previous Article Next Article

Research article

The generalized scale mixtures of asymmetric generalized normal distributions with application to stock data

Faculty of Science, Beijing University of Technology, Beijing 100124, China

Received: 11 August 2023 Revised: 23 November 2023 Accepted: 30 November 2023 Published: 07 December 2023
MSC : 62E15, 62F10, 62G30, 62P05

In this paper, we introduced a family of distributions with a very flexible shape named generalized scale mixtures of generalized asymmetric normal distributions (GSMAGN). We investigated the main properties of the new family including moments, skewness, kurtosis coefﬁcients and order statistics. A variant of the expectation maximization (EM)-type algorithm was established by combining the proflie likihood approach (PLA) with the classical expectation conditional maximization (ECM) algorithm for parameter estimation of this model. This approach with analytical expressions in the E-step and tractable M-step can greatly improve the computational speed and efficiency of the algorithm. The performance of the proposed algorithm was assessed by some simulation studies. The feasibility of the proposed methodology was illustrated through two real datasets.
- generalized asymmetric normal distribution,
- generalized scale mixtures,
- EM-type algorithm,
- order statistics
Citation: Ruijie Guan, Aidi Liu, Weihu Cheng. The generalized scale mixtures of asymmetric generalized normal distributions with application to stock data[J]. AIMS Mathematics, 2024, 9(1): 1291-1322. doi: 10.3934/math.2024064

Related Papers:

Abstract

In this paper, we introduced a family of distributions with a very flexible shape named generalized scale mixtures of generalized asymmetric normal distributions (GSMAGN). We investigated the main properties of the new family including moments, skewness, kurtosis coefﬁcients and order statistics. A variant of the expectation maximization (EM)-type algorithm was established by combining the proflie likihood approach (PLA) with the classical expectation conditional maximization (ECM) algorithm for parameter estimation of this model. This approach with analytical expressions in the E-step and tractable M-step can greatly improve the computational speed and efficiency of the algorithm. The performance of the proposed algorithm was assessed by some simulation studies. The feasibility of the proposed methodology was illustrated through two real datasets.

References

[1]	D. F. Andrews, C. L. Mallows, Scale mixtures of normal distribution, J. R. Stat. Soc. Ser. B Methodol., 36 (1974), 99–102. https://doi.org/10.1111/j.2517-6161.1974.tb00989.x doi: 10.1111/j.2517-6161.1974.tb00989.x
[2]	M. West, On scale mixtures of normal distributions, Biometrika, 74 (1987), 646–648. https://doi.org/10.1093/biomet/74.3.646 doi: 10.1093/biomet/74.3.646
[3]	M. D. Branco, D. K. Dey, A general class of multivariate skew-elliptical distributions, J. Multivar. Anal., 79 (2001), 99–113. https://doi.org/10.1006/jmva.2000.1960 doi: 10.1006/jmva.2000.1960
[4]	C. S. Ferreira, H. Bolfarine, V. H. Lachos, Linear mixed models based on skew scale mixtures of normal distribution, Commun. Stat. Simul. Comput., 51 (2020), 7194–7214. https://doi.org/10.1080/03610918.2020.1827265 doi: 10.1080/03610918.2020.1827265
[5]	R. M. Basso, V. H. Lachos, C. R. B. Cabral, P. Ghosh, Robust mixture modeling based on scale mixtures of skew-normal distributions, Comput. Stat. Data Anal., 54 (2010), 2926–2941. https://doi.org/10.1016/j.csda.2009.09.031 doi: 10.1016/j.csda.2009.09.031
[6]	H. M. Kim, M. G. Genton, Characteristic functions of scale mixtures of multivariate skew-normal distributions, Comput. Stat. Data Anal., 102 (2011), 1105–1117. https://doi.org/10.1016/j.jmva.2011.03.004 doi: 10.1016/j.jmva.2011.03.004
[7]	T. I. Lin, J. C. Lee, W. J. Hsieh, Robust mixture modeling using the skew t distribution, Stat. Comput., 17 (2007), 81–92. https://doi.org/10.1007/s11222-006-9005-8 doi: 10.1007/s11222-006-9005-8
[8]	T. I. Lin, H. J. Ho, C. R. Lee, Flexible mixture modelling using the multivariate skew-t-normal distribution, Stat. Comput., 24 (2014), 531–546. https://doi.org/10.1007/s11222-013-9386-4 doi: 10.1007/s11222-013-9386-4
[9]	I. Lin, J. C. Lee, Y. Y. Shu, Finite mixture modelling using the skew normal distribution, Stat. Sin., 17 (2007), 909–927. https://doi.org/10.2307/24307705 doi: 10.2307/24307705
[10]	A. Mahdavi, V. Amirzadeh, A. Jamalizadeh, T. I. Lin, Maximum likelihood estimation for scale-shape mixtures of flexible generalized skew normal distributions via selection representation, Comput. Stat., 36 (2021), 2201–2230. https://doi.org/10.1007/s00180-021-01079-2 doi: 10.1007/s00180-021-01079-2
[11]	A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Stat., 12 (1985), 171–178.
[12]	A. Azzalini, A. Dalla Valle, The multivariate skew-normal distribution, Biometrika, 83 (1996), 715–726. https://doi.org/10.1093/biomet/83.4.715 doi: 10.1093/biomet/83.4.715
[13]	A. Azzalini, A. Capitanio, Statistical applications of the multivariate skew normal distribution, J. R. Stat. Soc. Ser. B Methodol., 61 (1999), 579–602. https://doi.org/10.1111/1467-9868.00194 doi: 10.1111/1467-9868.00194
[14]	R. B. Arellano-Valle, A. Azzalini, On the unification of families of skew-normal distributions, Scand. J. Stat., 33 (2006), 561–574. https://doi.org/10.1111/j.1467-9469.2006.00503.x doi: 10.1111/j.1467-9469.2006.00503.x
[15]	R. B. Arellano-Valle, M. G. Genton, On fundamental skew distributions, J. Multivar. Anal., 96 (2005), 93–116. https://doi.org/10.1016/j.jmva.2004.10.002 doi: 10.1016/j.jmva.2004.10.002
[16]	A. Azzalini, The skew-normal distribution and related multivariate families, Scand. J. Stat., 32 (2005), 159–188. https://doi.org/10.1111/j.1467-9469.2005.00426.x doi: 10.1111/j.1467-9469.2005.00426.x
[17]	C. Fernandez, M. F. J. Steel, Reference priors for non-normal two-sample problems, Test, 7 (1988), 179–205. https://doi.org/10.1007/BF02565109 doi: 10.1007/BF02565109
[18]	D. M. Zhu, V. Zinde-Walsh, Properties and estimation of asymmetric exponential power distribution, J. Econom., 148 (2009), 86–99. https://doi.org/10.1016/j.jeconom.2008.09.038 doi: 10.1016/j.jeconom.2008.09.038
[19]	R. J. Guan, X. Zhao, C. H. Cheng, Y. H. Rong, A new generalized t distribution based on a distribution construction method, Mathematics, 9 (2021), 2413. https://doi.org/10.1016/10.3390/math9192413 doi: 10.1016/10.3390/math9192413
[20]	H. Exton, Handbook of hypergeometric integrals: Theory, applications, tables, computer programs, New York: Halsted Press, 1978.
[21]	A. P. Dempster, N. M. Laird, D. B. Rubin, Maximum likelihood from incomplete data via the em algorithm, J. R. Stat. Soc. Ser. B Methodol., 39 (1977), 1–22. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x doi: 10.1111/j.2517-6161.1977.tb01600.x
[22]	K. Lange, The EM algorithm, New York: Springer, 2013.
[23]	X. L. Meng, D. B. Rubin, Maximum likelihood estimation via the ecm algorithm: A general framework, Biometrika, 80 (1993), 267–278. https://doi.org/10.2307/2337198 doi: 10.2307/2337198
[24]	P. Huber, Robust statistics, New York: Wiley, 1981.
[25]	L. L. Wen, Y. J. Qiu, M. H. Wang, J. L. Yin, P. Y. Chen, Numerical characteristics and parameter estimation of finite mixed generalized normal distribution, Commun. Stat. Simul. Comput., 51 (2022), 3596–3620. https://doi.org/10.1080/03610918.2020.1720733 doi: 10.1080/03610918.2020.1720733

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)