Matrix variate generalized asymmetric Laplace distributions

Authors:
Tomasz J. Kozubowski, Stepan Mazur and Krzysztof Podgórski

Journal:
Theor. Probability and Math. Statist. **109** (2023), 55-80

MSC (2020):
Primary 62H10, 60E05; Secondary 60E10

DOI:
https://doi.org/10.1090/tpms/1197

Published electronically:
October 3, 2023

MathSciNet review:
4652994

Full-text PDF

Abstract |
References |
Similar Articles |
Additional Information

Abstract: The generalized asymmetric Laplace (GAL) distributions, also known as the variance/mean-gamma models, constitute a popular flexible class of distributions that can account for peakedness, skewness, and heavier-than-normal tails, often observed in financial or other empirical data. We consider extensions of the GAL distribution to the matrix variate case, which arise as covariance mixtures of matrix variate normal distributions. Two different mixing mechanisms connected with the nature of the random scaling matrix are considered, leading to what we term matrix variate GAL distributions of Type I and II. While Type I matrix variate GAL distribution has been studied before, there is no comprehensive account of Type II in the literature, except for their rather brief treatment as a special case of matrix variate generalized hyperbolic distributions. With this work we fill this gap, and present an account for basic distributional properties of Type II matrix variate GAL distributions. In particular, we derive their probability density function and the characteristic function, as well as provide stochastic representations related to matrix variate gamma distribution. We also show that this distribution is closed under linear transformations, and study the relevant marginal distributions. In addition, we also briefly account for Type I and discuss the intriguing connections with Type II. We hope that this work will be useful in the areas where matrix variate distributions provide an appropriate probabilistic tool for three-way or, more generally, panel data sets, which can arise across different applications.

References
- Laura Anderlucci and Cinzia Viroli,
*Covariance pattern mixture models for the analysis of multivariate heterogeneous longitudinal data*, Ann. Appl. Stat. **9** (2015), no. 2, 777–800. MR **3371335**, DOI 10.1214/15-AOAS816
- Anis Iranmanesh, M. Arashi, and S. M. M. Tabatabaey,
*On conditional applications of matrix variate normal distribution*, Iran. J. Math. Sci. Inform. **5** (2010), no. 2, 33–43 (English, with English and Persian summaries). MR **2866385**
- H. Asgharian, K. Podgórski, F. N. Shariati, and L. Liu,
*Structural Multivariate Spatial Econometrics: Application to Cross-Country Interdependence of Stock and Bond markets*, Available at SSRN: https://ssrn.com/abstract=3527346, 2018.
- O. Barndorff-Nielsen, J. Kent, and M. Sørensen,
*Normal variance-mean mixtures and $z$ distributions*, Internat. Statist. Rev. **50** (1982), no. 2, 145–159 (English, with French summary). MR **678296**, DOI 10.2307/1402598
- Ole E. Barndorff-Nielsen, Víctor Pérez-Abreu, and Alfonso Rocha-Arteaga,
*$\textrm {mat}\,G$ random matrices*, Stoch. Models **22** (2006), no. 4, 723–734. MR **2263863**, DOI 10.1080/15326340600878560
- A. Bekker and J. J. J. Roux,
*Bayesian multivariate normal analysis with a Wishart prior*, Comm. Statist. Theory Methods **24** (1995), no. 10, 2485–2497. MR **1354468**, DOI 10.1080/03610929508831629
- T. Bodnar, S. Mazur, S. Muhinyuza, and N. Parolya,
*On the product of a singular Wishart matrix and a singular Gaussian vector in high dimension*, Teor. Ĭmovīr. Mat. Stat. **99** (2018), 37–50 (English, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. **99** (2019), 39–52. MR **3908654**, DOI 10.1090/tpms/1078
- T. Bodnar, S. Mazur, and Y. Okhrin,
*Distribution of the product of singular Wishart matrix and normal vector*, Teor. Ĭmovīr. Mat. Stat. **91** (2014), 1–14; English transl., Theory Probab. Math. Statist. **91** (2015), 1–15. MR **3364119**, DOI 10.1090/tpms/962
- Taras Bodnar, Stepan Mazur, and Krzysztof Podgórski,
*Singular inverse Wishart distribution and its application to portfolio theory*, J. Multivariate Anal. **143** (2016), 314–326. MR **3431434**, DOI 10.1016/j.jmva.2015.09.021
- Y. Murat Bulut and Olcay Arslan,
*Matrix variate slash distribution*, J. Multivariate Anal. **137** (2015), 173–178. MR **3332805**, DOI 10.1016/j.jmva.2015.02.008
- A. P. Dawid,
*Some matrix-variate distribution theory: notational considerations and a Bayesian application*, Biometrika **68** (1981), no. 1, 265–274. MR **614963**, DOI 10.1093/biomet/68.1.265
- James M. Dickey,
*Matricvariate generalizations of the multivariate $t$ distribution and the inverted multivariate $t$ distribution*, Ann. Math. Statist. **38** (1967), 511–518. MR **208752**, DOI 10.1214/aoms/1177698967
- Jiu Ding and Aihui Zhou,
*Eigenvalues of rank-one updated matrices with some applications*, Appl. Math. Lett. **20** (2007), no. 12, 1223–1226. MR **2384251**, DOI 10.1016/j.aml.2006.11.016
- Peng Ding and Joseph K. Blitzstein,
*On the Gaussian mixture representation of the Laplace distribution*, Amer. Statist. **72** (2018), no. 2, 172–174. MR **3810626**, DOI 10.1080/00031305.2017.1291448
- F. Z. Doğru, Y. M. Bulut, and O. Arslan,
*Finite mixtures of matrix variate t distributions*, Gazi University Journal of Science **29** (2016), no. 2, 335–341.
- Michael P. B. Gallaugher and Paul D. McNicholas,
*A matrix variate skew-$t$ distribution*, Stat **6** (2017), 160–170. MR **3653050**, DOI 10.1002/sta4.143
- —,
*Finite mixtures of skewed matrix variate distributions*, Pattern Recognition **80** (2018), 83–93.
- Michael P. B. Gallaugher and Paul D. McNicholas,
*Three skewed matrix variate distributions*, Statist. Probab. Lett. **145** (2019), 103–109. MR **3873895**, DOI 10.1016/j.spl.2018.08.012
- Michael P. B. Gallaugher and Paul D. McNicholas,
*Mixtures of skewed matrix variate bilinear factor analyzers*, Adv. Data Anal. Classif. **14** (2020), no. 2, 415–434. MR **4118954**, DOI 10.1007/s11634-019-00377-4
- Seymour Geisser,
*Bayesian estimation in multivariate analysis*, Ann. Math. Statist. **36** (1965), 150–159. MR **174134**, DOI 10.1214/aoms/1177700279
- A. K. Gupta and D. K. Nagar,
*Matrix variate distributions*, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 104, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR **1738933**
- A. K. Gupta and T. Varga,
*Normal mixture representations of matrix variate elliptically contoured distributions*, Sankhyā Ser. A **57** (1995), no. 1, 68–78. MR **1392632**
- Solomon W. Harrar, Eugene Seneta, and Arjun K. Gupta,
*Duality between matrix variate $t$ and matrix variate V.G. distributions*, J. Multivariate Anal. **97** (2006), no. 6, 1467–1475. MR **2256162**, DOI 10.1016/j.jmva.2005.09.002
- David A. Harville,
*Matrix algebra from a statistician’s perspective*, Springer-Verlag, New York, 1997. MR **1467237**, DOI 10.1007/b98818
- Carl S. Herz,
*Bessel functions of matrix argument*, Ann. of Math. (2) **61** (1955), 474–523. MR **69960**, DOI 10.2307/1969810
- Md. Mobarak Hossain, Tomasz J. Kozubowski, and Krzysztof Podgórski,
*A novel weighted likelihood estimation with empirical Bayes flavor*, Comm. Statist. Simulation Comput. **47** (2018), no. 2, 392–412. MR **3757694**, DOI 10.1080/03610918.2016.1197246
- B. M. Golam Kibria,
*The matrix-$t$ distribution and its applications in predictive inference*, J. Multivariate Anal. **97** (2006), no. 3, 785–795. MR **2236502**, DOI 10.1016/j.jmva.2005.08.001
- Samuel Kotz, Tomasz J. Kozubowski, and Krzysztof Podgórski,
*The Laplace distribution and generalizations*, Birkhäuser Boston, Inc., Boston, MA, 2001. A revisit with applications to communications, economics, engineering, and finance. MR **1935481**, DOI 10.1007/978-1-4612-0173-1
- T. J. Kozubowski, S. Mazur, and K. Podgórski,
*Matrix gamma distributions and related stochastic processes*, Working Paper 12, Örebro University, 2022.
- Tomasz J. Kozubowski and Krzysztof Podgórski,
*Gaussian mixture representation of the Laplace distribution revisited: bibliographical connections and extensions*, Amer. Statist. **74** (2020), no. 4, 407–412. MR **4168260**, DOI 10.1080/00031305.2019.1630000
- Tomasz J. Kozubowski, Krzysztof Podgórski, and Igor Rychlik,
*Multivariate generalized Laplace distribution and related random fields*, J. Multivariate Anal. **113** (2013), 59–72. MR **2984356**, DOI 10.1016/j.jmva.2012.02.010
- D. B. Madan and E. Seneta,
*The variance gamma (V.G.) model for share market returns*, The Journal of Business **63** (1990), no. 4, 511–524.
- Pierre-Alexandre Mattei,
*Multiplying a Gaussian matrix by a Gaussian vector*, Statist. Probab. Lett. **128** (2017), 67–70. MR **3656377**, DOI 10.1016/j.spl.2017.04.004
- Volodymyr Melnykov and Xuwen Zhu,
*On model-based clustering of skewed matrix data*, J. Multivariate Anal. **167** (2018), 181–194. MR **3830641**, DOI 10.1016/j.jmva.2018.04.007
- Anne Opschoor, Pawel Janus, André Lucas, and Dick Van Dijk,
*New HEAVY models for fat-tailed realized covariances and returns*, J. Bus. Econom. Statist. **36** (2018), no. 4, 643–657. MR **3871707**, DOI 10.1080/07350015.2016.1245622
- Victor Pérez-Abreu and Robert Stelzer,
*Infinitely divisible multivariate and matrix gamma distributions*, J. Multivariate Anal. **130** (2014), 155–175. MR **3229530**, DOI 10.1016/j.jmva.2014.04.017
- L. Thabane and M. Safiul Haq,
*On the matrix-variate generalized hyperbolic distribution and its Bayesian applications*, Statistics **38** (2004), no. 6, 511–526. MR **2109632**, DOI 10.1080/02331880412331319279
- Geoffrey Z. Thompson, Ranjan Maitra, William Q. Meeker, and Ashraf F. Bastawros,
*Classification with the matrix-variate-$t$ distribution*, J. Comput. Graph. Statist. **29** (2020), no. 3, 668–674. MR **4153190**, DOI 10.1080/10618600.2019.1696208
- Cinzia Viroli,
*Finite mixtures of matrix normal distributions for classifying three-way data*, Stat. Comput. **21** (2011), no. 4, 511–522. MR **2826689**, DOI 10.1007/s11222-010-9188-x
- Cinzia Viroli,
*Model based clustering for three-way data structures*, Bayesian Anal. **6** (2011), no. 4, 573–602. MR **2869958**, DOI 10.1214/11-BA622
- Y. Yurchenko,
*Matrix variate and tensor variate Laplace distributions*, Available at arXiv: https://arxiv.org/abs/2104.05669, 2021.

References
- L. Anderlucci and C. Viroli,
*Covariance pattern mixture models for the analysis of multivariate heterogeneous longitudinal data*, Ann. Appl. Stat. **9** (2015), no. 2, 777–800. MR **3371335**
- I. Anis, M. Arashi, and S. M. M. Tabatabaey,
*On conditional applications of matrix variate normal distribution*, Iran. J. Math. Sci. Inform. **5** (2010), no. 2, 33–43. MR **2866385**
- H. Asgharian, K. Podgórski, F. N. Shariati, and L. Liu,
*Structural Multivariate Spatial Econometrics: Application to Cross-Country Interdependence of Stock and Bond markets*, Available at SSRN: https://ssrn.com/abstract=3527346, 2018.
- O. E. Barndorff-Nielsen, J. Kent, and M. Sørensen,
*Normal variance-mean mixtures and z distributions*, Internat. Statist. Rev. **50** (1982), no. 2, 145–159. MR **678296**
- O. E. Barndorff-Nielsen, V. Pérez-Abreu, and A. Rocha-Arteaga,
*Mat$G$ random matrices*, Stoch. Models **22** (2006), no. 4, 723–734. MR **2263863**
- A. Bekker and J. J. J. Roux,
*Bayesian multivariate normal analysis with a Wishart prior*, Comm. Statist. Theory Methods **24** (1995), no. 10, 2485–2497. MR **1354468**
- T. Bodnar, S. Mazur, S. Muhinyuza, and N. Parolya,
*On the product of a singular Wishart matrix and a singular Gaussian vector in high dimension*, Theory Probab. Math. Statist. **99** (2018), no. 2, 37–50. MR **3908654**
- T. Bodnar, S. Mazur, and Y. Okhrin,
*Distribution of the product of singular Wishart matrix and normal vector*, Theory Probab. Math. Statist **91** (2014), 1–15. MR **3364119**
- T. Bodnar, S. Mazur, and K. Podgórski,
*Singular inverse Wishart distribution and its application to portfolio theory*, J. Multivariate Anal. **143** (2016), 314–326. MR **3431434**
- Y. M. Bulut and O. Arslan,
*Matrix variate slash distribution*, J. Multivariate Anal. **137** (2015), 173–178. MR **3332805**
- A. P. Dawid,
*Some matrix-variate distribution theory: Notational considerations and a Bayesian application*, Biometrika **68** (1981), no. 1, 265–274. MR **614963**
- J. M. Dickey,
*Matricvariate generalizations of the multivariate $t$ distribution and the inverted multivariate $t$ distribution*, Ann. Math. Statist. **38** (1967), no. 2, 511–518. MR **208752**
- J. Ding and A. Zhou,
*Eigenvalues of rank-one updated matrices with some applications*, Appl. Math. Lett. **20** (2007), no. 12, 1223–1226. MR **2384251**
- P. Ding and J. K. Blitzstein,
*On the Gaussian mixture representation of the Laplace distribution*, Amer. Statist. **72** (2018), no. 2, 172–174. MR **3810626**
- F. Z. Doğru, Y. M. Bulut, and O. Arslan,
*Finite mixtures of matrix variate t distributions*, Gazi University Journal of Science **29** (2016), no. 2, 335–341.
- M. P. B. Gallaugher and P. D. McNicholas,
*A matrix variate skew-$t$ distribution*, Stat **6** (2017), no. 1, 160–170. MR **3653050**
- —,
*Finite mixtures of skewed matrix variate distributions*, Pattern Recognition **80** (2018), 83–93.
- —,
*Three skewed matrix variate distributions*, Statist. Probab. Lett. **145** (2019), 103–109. MR **3873895**
- —,
*Mixtures of skewed matrix variate bilinear factor analyzers*, Adv. Data Anal. Classif. **14** (2020), no. 2, 415–434. MR **4118954**
- S. Geisser,
*Bayesian estimation in multivariate analysis*, Ann. Math. Statist. **36** (1965), 150–159. MR **174134**
- A. K. Gupta and D. K. Nagar,
*Matrix variate distributions*, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR **1738933**
- A. K. Gupta and T. Varga,
*Normal mixture representations of matrix variate elliptically contoured distributions*, Sankhyā Ser. A **57** (1995), no. 1, 68–78. MR **1392632**
- S. W. Harrar, E. Seneta, and A. K. Gupta,
*Duality between matrix variate t and matrix variate V.G. distributions*, J. Multivariate Anal. **97** (2006), no. 6, 1467–1475. MR **2256162**
- D. A. Harville,
*Matrix algebra from a statistician perspective*, Springer, 1997. MR **1467237**
- C. S. Herz,
*Bessel functions of matrix argument*, Ann. of Math. **61** (1955), no. 3, 474–523. MR **69960**
- Md. M. Hossain, T. J. Kozubowski, and K. Podgórski,
*A novel weighted likelihood estimation with empirical bayes flavor*, Comm. Statist. Simulation Comput. **47** (2018), no. 2, 392–412. MR **3757694**
- G. B. M. Kibria,
*The matrix-t distribution and its applications in predictive inference*, J. Multivariate Anal. **97** (2006), no. 3, 785–795. MR **2236502**
- S. Kotz, T. J. Kozubowski, and K. Podgórski,
*The Laplace distribution and generalizations: A revisit with applications to communications, economics, engineering and finance*, Birkhäuser Boston, Inc., Boston, MA, 2001. MR **1935481**
- T. J. Kozubowski, S. Mazur, and K. Podgórski,
*Matrix gamma distributions and related stochastic processes*, Working Paper 12, Örebro University, 2022.
- T. J. Kozubowski and K. Podgórski,
*Gaussian mixture representation of the Laplace distribution revisited: Bibliographical connections and extensions*, Amer. Statist. **74** (2020), no. 4, 407–412. MR **4168260**
- T. J. Kozubowski, K. Podgórski, and I. Rychlik,
*Multivariate generalized Laplace distribution and related random fields*, J. Multivariate Anal. **113** (2013), 59–72, Special Issue on Multivariate Distribution Theory in Memory of Samuel Kotz. MR **2984356**
- D. B. Madan and E. Seneta,
*The variance gamma (V.G.) model for share market returns*, The Journal of Business **63** (1990), no. 4, 511–524.
- P.-A. Mattei,
*Multiplying a Gaussian matrix by a Gaussian vector*, Statist. Probab. Lett. **128** (2017), 67–70. MR **3656377**
- V. Melnykov and X. Zhu,
*On model-based clustering of skewed matrix data*, J. Multivariate Anal. **167** (2018), 181–194. MR **3830641**
- A. Opschoor, P. Janus, A. Lucas, and D. Van Dijk,
*New heavy models for fat-tailed realized covariances and returns*, J. Bus. Econom. Statist. **36** (2018), no. 4, 643–657. MR **3871707**
- V. Pérez-Abreu and R. Stelzer,
*Infinitely divisible multivariate and matrix gamma distributions*, J. Multivariate Anal. **130** (2014), 155–175. MR **3229530**
- L. Thabane and M. Safiul Haq,
*On the matrix-variate generalized hyperbolic distribution and its Bayesian applications*, Statistics **38** (2004), no. 6, 511–526. MR **2109632**
- G. Z. Thompson, R. Maitra, W. Q. Meeker, and A. F. Bastawros,
*Classification with the matrix-variate-t distribution*, J. Comput. Graph. Statist. **29** (2020), no. 3, 668–674. MR **4153190**
- C. Viroli,
*Finite mixtures of matrix normal distributions for classifying three-way data*, Stat. Comput. **21** (2011), no. 4, 511–522. MR **2826689**
- —,
*Model based clustering for three-way data structures*, Bayesian Anal. **6** (2011), no. 4, 573–602. MR **2869958**
- Y. Yurchenko,
*Matrix variate and tensor variate Laplace distributions*, Available at arXiv: https://arxiv.org/abs/2104.05669, 2021.

Similar Articles

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2020):
62H10,
60E05,
60E10

Retrieve articles in all journals
with MSC (2020):
62H10,
60E05,
60E10

Additional Information

**Tomasz J. Kozubowski**

Affiliation:
Department of Mathematics and Statistics, University of Nevada, NV-89557 Reno

Email:
tkozubow@unr.edu

**Stepan Mazur**

Affiliation:
Department of Statistics, Örebro University, SE-70182 Örebro, Sweden; and Department of Economics and Statistics, Linnaeus University, SE-35195 Växjö, Sweden

Email:
Stepan.Mazur@oru.se

**Krzysztof Podgórski**

Affiliation:
Department of Statistics, Lund University, SE-22007 Lund, Sweden

Email:
Krzysztof.Podgorski@stat.lu.se

Keywords:
Covariance mixture of Gaussian distributions,
distribution theory,
generalized asymmetric Laplace distribution,
MatG distribution,
matrix variate distribution,
matrix variate gamma distribution,
matrix gamma-normal distribution,
matrix variate $t$ distribution,
normal variance-mean mixture,
variance gamma distribution

Received by editor(s):
June 6, 2022

Accepted for publication:
January 4, 2023

Published electronically:
October 3, 2023

Additional Notes:
The second author acknowledges financial support from the internal research grants at Örebro University and from the project “Models for macro and financial economics after the financial crisis” (Dnr: P18-0201) funded by Jan Wallander and Tom Hedelius foundation.

The third author acknowledges financial support of the Swedish Research Council (VR) Grant DNR: 2020-05168.

Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv