Integral stochastic ordering of the multivariate normal mean-variance and the skew-normal scale-shape mixture models
Abstract
In this paper, we introduce integral stochastic ordering of two most important classes of distributions that are commonly used to fit data possessing high values of skewness and (or) kurtosis. The first one is based on the selection distributions started by the univariate skew-normal distribution. A broad, flexible and newest class in this area is the scale and shape mixture of multivariate skew-normal distributions. The second one is the general class of Normal Mean-Variance Mixture distributions. We then derive necessary and sufficient conditions for comparing the random vectors from these two classes of distributions. The integral orders considered here are the usual, concordance, supermodular, convex, increasing convex and directionally convex stochastic orders. Moreover, for bivariate random vectors, in the sense of stop-loss and bivariate concordance stochastic orders, the dependence strength of random portfolios is characterized in terms of order of correlations.References
R. B. Arellano-Valle, C. S. Ferreira, and M. G. Genton, Scale and shape mixtures of multivariate skew-normal distributions, Journal of Multivariate Analysis, vol. 166, pp. 98–110, 2018.
R. B. Arellano-Valle, M. G. Genton, and R. H. Loschi, Shape mixtures of multivariate skew-normal distributions, Journal of Multivariate Analysis, vol. 100, no. 1, pp. 91–101, 2009.
R. B. Arellano-Valle, H. W. G´omez, and F. A. Quintana, A new class of skew-normal distributions, Communications in Statistics-Theory and Methods, vol. 33, no. 7, pp. 1465–1480, 2004.
O. Arslan, An alternative multivariate skew-slash distribution, Statistics and Probability Letters, vol. 78, no. 16, pp. 2756–2761,2008.
A. Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, vol. 12, pp. 171–178, 1985.
A. Azzalini, and A. Capitanio, Statistical applications of the multivariate skew-normal distribution, Journal of the Royal Statistical Society: Series B, vol. 61, no. 3, pp. 579–602, 1999.
A. Azzalini, and A. Capitanio, Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution, Journal of the Royal Statistical Society, Series B, vol. 65, pp. 367–389, 2003.
A. Azzalini, and A. Dalla Valle, The multivariate skew-normal distribution, Biometrika, vol. 83, pp. 715–726, 1996.
A. Azzalini, and G. Regoli, Some properties of skew-symmetric distributions, Annals of the Institute of Statistical Mathematics, vol.64, no. 4, pp. 857–879, 2012.
O. Barndorff-Nielsen, and P. Blaesild, Hyperbolic distributions and ramifications: Contributions to theory and application, In Statistical Distributions in Scientific Work, Eds. C. Taillie, G. P. Patil and B. Baldessari, Dordrecht: Reidel, pp. 19–44, 1981.
N. B¨auerle, Inequalities for stochastic models via supermodular orderings, Stochastic Models, vol. 13, no. 1, pp. 181–201, 1997.
J. Behboodian, N. Balakrishnan, and A. Jamalizadeh, A new class of skew-Cauchy distributions, Statistics and Probability Letters, vol. 76, pp. 1488–1493, 2006.
F. Belzunce, An introduction to the theory of stochastic orders, Bolet´ın de Estad´ıstica e Investigaci´on Operativa (SEIO), vol. 26, no.1, pp. 4–18, 2010.
B. M. Bibby, and M. Sorensen, Hyperbolic processes in finance, in: Handbook of Heavy Tailed Distributions in Finance(Ed., S.T.Rachev), North-Holland, Amsterdam, pp. 211–248, 2003.
M. D. Branco, and D. K. Dey, A general class of multivariate skew-elliptical distributions, Journal of Multivariate Analysis, vol.79, pp. 99–113, 2001.
R. Cont, and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall, London, 2004.
O. Davidov, and S. Peddada, The linear stochastic order and directed inference for multivariate ordered distributions, The Annals of Statistics, vol. 41, pp. 1–40, 2013.
M. Denuit, and A. M¨uller, Smooth generators of integral stochastic orders, The Annals of Applied Probability, vol. 12, no. 4, pp.1174–1184, 2002.
J. Dhaene, and M. Goovaerts, Dependency of risks and stop-loss order, ASTIN Bulletin, vol. 26, no. 2, pp. 201–212, 1996.
C. S. Ferreira, V. H. Lachos, and H. Bolfarine, Likelihood-based inference for multivariate skew scale mixtures of normal distributions, AStA Advances in Statistical Analysis, vol. 100, pp. 421–441, 2016.
M. G. Genton, Skew-elliptical Distributions and Their Applications: A Journey Beyond Normality, Edited volume, Chapman & Hall/CRC Press, Boca Raton, 2004.
H. W. G´omez, O. Venegas, and H. Bolfarine, Skew-symmetric distributions generated by the distribution function of the normal distribution, Environmetrics, vol. 18, pp. 395–407, 2007.
R. C. Gupta, and R. D. Gupta, Generalized skew-normal model, Test, vol. 13, no. 2, pp. 501–524, 2004.
W. H¨urliman, On likelihood ratio and stochastic order for skew-symmetric distributions with a common kernel, International Journal of Contemporary Mathematical Sciences, vol. 8, no. 20, pp. 957–967, 2013.
D. Jamali, M. Amiri, and A. Jamalizadeh, Comparison of the multivariate skew-normal random vectors based on the integral stochastic ordering, Communications in Statistics-Theory and Methods, (To appear), 2020.
A. Jamalizadeh, and T. I. Lin, A general class of scale-shape mixtures of skew-normal distributions: properties and estimation,Computational Statistics, vol. 32, no. 2, pp. 451–474, 2017.
F. Kahrari, M. Rezaei, F. Yousefzadeh, and R. B. Arellano-Valle, On the multivariate skew-normal-Cauchy distribution, Statistics and Probability Letters, vol. 17, pp. 80–88, 2016.
H. M. Kim, and M. G. Genton, Characteristic functions of scale mixtures of multivariate skew-normal distributions, Journal of Multivariate Analysis, vol. 102, no. 7, pp. 1105–1117, 2011.
Z. Landsman, and A. Tsanakas, Stochastic ordering of bivariate elliptical distributions, Statistics and Probability Letters, vol. 76,pp. 488–494, 2006.
V. H. Lachos, D. K. Dey, and V. G. Cancho, Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective, Journal of Statistical Planning and Inference, vol. 139, pp. 4098–4110, 2006.
A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princeton university press, Princeton, 2005.
K. Mosler, Characterization of some stochastic orderings in multinormal and elliptic distributions, Operations Research, Proceeding of the 12-th Annual Meeting in Mannheim, pp. 520–527, 1983.
A. M¨uller, Stochastic orders generated by integrals: a unified study, Advances in Applied Probability, vol. 29, no. 2, pp. 414–428,1997.
A. M¨uller, Stochastic ordering of multivariate normal distributions, Annals of the Institute of Statistical Mathematics, vol. 53, no.3, pp. 567–575, 2001.
A. M¨uller, and D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley, New York, 2002.
M. Naderi, A. Arabpour, and A. Jamalizadeh, Multivariate normal mean-variance mixture distribution based on Lindley distribution,Communications in Statistics-Simulation and Computation, vol. 47, no. 4, pp. 1179–1192, 2018.
R. B. Nelsen, An Introduction to Copulas, Springer, New York, 1999.
X. Pan, G. Qiu, and T. Hu, Stochastic orderings for elliptical random vectors, Journal of Multivariate Analysis, vol. 148, pp. 83–88,2016.
R. Pourmousa, A. Jamalizadeh, and M. Rezapour, Multivariate normal mean-variance mixture distribution based on Birnbaum-Saunders distribution, Journal of Statistical Computation and Simulation, vol. 85, no. 13, pp. 2736–2749, 2015.
M. Rothschild, and J. Stiglitz, Increasing Risk: I. A Deflnition, Journal of Economic Theory, vol. 2, pp. 225–243, 1970.
L. R¨uschendorf, Inequalities for the expectation of Δ-monotone functions, Z. Wahrsch. Verw. Gebiete, vol. 54, pp. 341–349, 1980.
T. H. Rydberg, Generalized hyperbolic diffusion processes with applications in finance, Mathematical Finance, vol. 9, pp. 183–201,1999.
M. Scarsini, Multivariate convex orderings, dependence, and stochastic equality, Journal of Applied Probability, vol. 35, no. 1, pp.93–103, 1998.
M. Shaked, and J. G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, Boston, 1994.
M. Shaked, and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007.
D. Slepian, The one sided barrier problem for Gaussian noise, Bell System Technical Journal, vol. 41, no. 2, pp. 463–501, 1962.
J. Wang, and M. G. Genton, The multivariate skew-slash distribution, Journal of Statistical Planning and Inference, vol. 136, pp.209–220, 2006.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
