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A formulation for continuous mixtures of multivariate normal distributions. (English) Zbl 1476.62097

Summary: Several formulations have long existed in the literature in the form of continuous mixtures of normal variables where a mixing variable operates on the mean or on the variance or on both the mean and the variance of a multivariate normal variable, by changing the nature of these basic constituents from constants to random quantities. More recently, other mixture-type constructions have been introduced, where the core random component, on which the mixing operation operates, is not necessarily normal. The main aim of the present work is to show that many existing constructions can be encompassed by a formulation where normal variables are mixed using two univariate random variables. For this formulation, we derive various general properties, with focus on the multivariate context. Within the proposed framework, it is also simpler to formulate new proposals of parametric families, and we provide a few such instances. As a side product, the exposition provides a concise compendium of the main constructions of continuous normal-mixtures type, although a full overview of this vast theme is not attempted.

MSC:

62H10 Multivariate distribution of statistics
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E05 Probability distributions: general theory

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References:

[1] Adcock, C. J., Asset pricing and portfolio selection based on the multivariate extended skew-Student-\(t\) distribution, Ann. Oper. Res., 176, 221-234 (2010) · Zbl 1233.91112
[2] Adcock, C. J.; Shutes, K., On the multivariate extended skew-normal, normal-exponential, and normal-gamma distributions, J. Stat. Theory Pract., 6, 636-664 (2012) · Zbl 1425.62027
[3] Andrews, D. F.; Mallows, C. L., Scale mixtures of normal distributions, J. R. Stat. Soc. Ser. B Stat. Methodol., 36, 99-102 (1974) · Zbl 0282.62017
[4] Arellano-Valle, R. B.; Azzalini, A., On the unification of families of skew-normal distributions, Scand. J. Stat., 33, 561-574 (2006) · Zbl 1117.62051
[5] Arellano-Valle, R. B.; Azzalini, A.; Ferreira, C. S.; Santoro, K., A two-piece normal measurement error model, Comp. Statist. Data An., 144 (2021), Available online 14 2019
[6] Arellano-Valle, R. B.; Bolfarine, H.; Lachos, V. H., Skew-normal linear mixed models, J. Data Sci., 3, 415-438 (2005)
[7] Arellano-Valle, R. B.; Genton, M. G., Multivariate extended skew-\(t\) distributions and related families, (Metron LXVIII (2010)), 201-234 · Zbl 1301.62016
[8] Arellano-Valle, R. B.; Ozán, S.; Bolfarine, H.; Lachos, V. H., Skew-normal measurement error models, J. Multivariate Anal., 96, 265-281 (2005) · Zbl 1077.62043
[9] Arslan, O., Variance-mean mixture of the multivariate skew normal distribution, Statist. Pap., 56, 353-378 (2015) · Zbl 1309.62043
[10] Azzalini, A.; Capitanio, A., (The Skew-Normal and Related Families. The Skew-Normal and Related Families, IMS monographs (2014), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0924.62050
[11] Azzalini, A.; Dalla Valle, A., The multivariate skew-normal distribution, Biometrika, 83, 715-726 (1996) · Zbl 0885.62062
[12] Barndorff-Nielsen, O., Exponentially decreasing distributions for logarithm of particle size, Proc. Roy. Soc. London, Ser. A, 353, 401-419 (1977)
[13] Barndorff-Nielsen, O., Hyperbolic distributions and distributions on hyperbolae, Scand. J. Stat., 5, 151-157 (1978) · Zbl 0386.60018
[14] Blæsild, P., The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen’s Bean Data, Biometrika, 68, 251-263 (1981) · Zbl 0463.62048
[15] Branco, M. D.; Dey, D. K., A general class of multivariate skew-elliptical distributions, J. Multivariate Anal., 79, 99-113 (2001) · Zbl 0992.62047
[16] DeGroot, M. H., Optimal Statistical Decisions (1970), McGraw-Hill: McGraw-Hill New York · Zbl 0225.62006
[17] Elandt, R. C., The folded normal distribution: two methods of estimating parameters from moment, Technometrics, 3, 551-562 (1961) · Zbl 0103.37006
[18] Fang, K.-T.; Kotz, S.; Ng, K. W., Symmetric Multivariate and Related Distributions (1990), Chapman & Hall, London · Zbl 0699.62048
[19] Forbes, F.; Wraith, D., A new family of multivariate heavy-tailed distributions with variable marginal amounts of tailweight: application to robust clustering, Stat. Comput., 24, 971-984 (2014) · Zbl 1332.62204
[20] Gómez-Sánchez-Manzano, E.; Gómez-Villegas, M. A.; Marín, J. M., Multivariate exponential power distributions as mixtures of normal distributions with Bayesian applications, Commun. Statist. Theory Methods, 37, 972-985 (2008) · Zbl 1135.62041
[21] Jørgensen, B., (Statistical Properties of the Generalized Inverse Gaussian Distribution. Statistical Properties of the Generalized Inverse Gaussian Distribution, Lecture Notes in Statistics, vol. 9 (1982), Springer-Verlag) · Zbl 0486.62022
[22] Kano, Y., Consistency property of elliptical probability density functions, J. Multivariate Anal., 51, 139-147 (1994) · Zbl 0806.62039
[23] Kotz, S.; Nadarajah, S., Multivariate \(t\) Distributions and their Applications (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1100.62059
[24] Lange, K.; Sinsheimer, J. S., Normal/independent distributions and their applications in robust regression, J. Comput. Graph. Stat., 2, 175-198 (1993)
[25] Madan, D. B.; Seneta, E., The variance gamma (V. G.) model for share market returns, J. Bus., 63, 511-524 (1990)
[26] Mardia, K., Measures of multivariate skewness and kurtosis with applications, Biometrika, 57, 519-530 (1970) · Zbl 0214.46302
[27] Mardia, K. V., Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies, Sankhyā, Ser. B, 36, 115-128 (1974) · Zbl 0345.62031
[28] McNeil, A. J.; Frey, R.; Embrechts, P., (Quantitative Risk Management : Concepts, Techniques and Tools. Quantitative Risk Management : Concepts, Techniques and Tools, Princeton Series in Finance (2005), Princeton University Press) · Zbl 1089.91037
[29] Nakagami, M., The \(m\)-distribution—a general formula of intensity distribution of rapid fading, (Hoffman, W. C., Statistical Methods in Radio Wave Propagation (1960), Permamon Press), 3-6, 6a, 7-36, Proceedings of a Symposium Held At the University of California, Los Angeles, June 18-20 (1958)
[30] Negarestani, H.; Jamalizadeh, A.; Shafiei, S.; Balakrishnan, N., Mean mixtures of normal distributions: properties, inference and application, Metrika, 82, 501-528 (2019) · Zbl 1481.60026
[31] Nelson, V., Wind Energy: Renewable Energy and the Environment (2013), CRC Press
[32] Paolella, M. S., Intermediate Probability: A Computational Approach (2007), J. Wiley & Sons: J. Wiley & Sons New York · Zbl 1149.60002
[33] Sichel, H. S., Statistical valuation of diamondiferous deposits, J. S. Afr. Inst. Min. Metall., 73, 235-243 (1973)
[34] Simaan, Y., Portfolio selection and asset pricing-three-parameter framework, Manage. Sci., 39, 568-577 (1993) · Zbl 0783.90012
[35] Subbotin, M. T., On the law of frequency of error, Mat. Sb., 31, 296-301 (1923) · JFM 49.0370.01
[36] Tjetjep, A.; Seneta, E., Skewed normal variance-mean models for asset pricing and the method of moments, Int. Statist. Rev., 74, 109-126 (2006) · Zbl 1131.62096
[37] Vilca, F.; Balakrishnan, N.; Zeller, C. B., Multivariate skew-normal generalized hyperbolic distribution and its properties, J. Multivariate Anal., 128, 73-85 (2014) · Zbl 1352.62080
[38] Wallis, K. F., The two-piece normal binormal, or double Gaussian distribution: its origin and rediscoveries, Stat. Sci., 29, 106-112 (2014) · Zbl 1332.60009
[39] Wang, J.; Genton, M. G., The multivariate skew-slash distribution, J. Statist. Plann. Inference, 136, 209-220 (2006) · Zbl 1081.60013
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