IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v98y2007i8p1684-1692.html
   My bibliography  Save this article

Pseudo-inverse multivariate/matrix-variate distributions

Author

Listed:
  • Zhang, Zhihua

Abstract

The Moore-Penrose inverse of a singular or nonsquare matrix is not only existent but also unique. In this paper, we derive the Jacobian of the transformation from such a matrix to the transpose of its Moore-Penrose inverse. Using this Jacobian, we investigate the distribution of the Moore-Penrose inverse of a random matrix and propose the notion of pseudo-inverse multivariate/matrix-variate distributions. For arbitrary multivariate or matrix-variate distributions, we can develop the corresponding pseudo-inverse distributions. In particular, we present pseudo-inverse multivariate normal distributions, pseudo-inverse Dirichlet distributions, pseudo-inverse matrix-variate normal distributions and pseudo-inverse Wishart distributions.

Suggested Citation

  • Zhang, Zhihua, 2007. "Pseudo-inverse multivariate/matrix-variate distributions," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1684-1692, September.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:8:p:1684-1692
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(06)00050-9
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. José Díaz-García & Ramón Gutiérrez-Jáimez, 2005. "Functions of singular random matrices with applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(2), pages 475-487, December.
    2. Díaz-García, José A. & Jáimez, Ramón Gutierrez & Mardia, Kanti V., 1997. "Wishart and Pseudo-Wishart Distributions and Some Applications to Shape Theory," Journal of Multivariate Analysis, Elsevier, vol. 63(1), pages 73-87, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Omer L. Gebizlioglu & Serap Yörübulut, 2016. "A Pseudo-Pareto Distribution and Concomitants of Its Order Statistics," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 1043-1064, December.
    2. Kjetil B. Halvorsen & Victor Ayala & Eduardo Fierro, 2016. "On the Marginal Distribution of the Diagonal Blocks in a Blocked Wishart Random Matrix," International Journal of Analysis, Hindawi, vol. 2016, pages 1-5, November.
    3. José Díaz-García & Francisco Caro-Lopera, 2012. "Generalised shape theory via SV decomposition I," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(4), pages 541-565, May.
    4. Díaz García, José A. & González Farías, Graciela, 2002. "Singular random matrix decompositions: distributions," DES - Working Papers. Statistics and Econometrics. WS ws024211, Universidad Carlos III de Madrid. Departamento de Estadística.
    5. Díaz-García, José A. & Gutiérrez-Jáimez, Ramón, 2006. "The distribution of the residual from a general elliptical multivariate linear model," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1829-1841, September.
    6. Tsukuma, Hisayuki, 2016. "Estimation of a high-dimensional covariance matrix with the Stein loss," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 1-17.
    7. Bodnar, Taras & Okhrin, Yarema, 2008. "Properties of the singular, inverse and generalized inverse partitioned Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2389-2405, November.
    8. Liu, Jin Shan & Ip, Wai Cheung & Wong, Heung, 2009. "Predictive inference for singular multivariate elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1440-1446, August.
    9. Díaz-García, José A., 2007. "A note about measures and Jacobians of singular random matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 960-969, May.
    10. Bodnar, Taras & Mazur, Stepan & Podgórski, Krzysztof, 2016. "Singular inverse Wishart distribution and its application to portfolio theory," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 314-326.
    11. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2015. "Estimation of the mean vector in a singular multivariate normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 245-258.
    12. Díaz García, José A. & González Farías, Graciela, 2002. "Singular random matrix decompositions: Jacobians," DES - Working Papers. Statistics and Econometrics. WS ws024110, Universidad Carlos III de Madrid. Departamento de Estadística.
    13. Hisayuki Tsukuma & Tatsuya Kubokawa, 2014. "Estimation and Prediction Intervals in Transformed Linear Mixed Models," CIRJE F-Series CIRJE-F-930, CIRJE, Faculty of Economics, University of Tokyo.
    14. Díaz-García, José A. & González-Farías, Graciela, 2005. "Singular random matrix decompositions: distributions," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 109-122, May.
    15. Shimizu, Koki & Hashiguchi, Hiroki, 2021. "Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    16. Andrea Carriero & Francesco Corsello & Massimiliano Marcellino, 2020. "The economic drivers of volatility and uncertainty," Temi di discussione (Economic working papers) 1285, Bank of Italy, Economic Research and International Relations Area.
    17. Díaz-García, José A. & González-Farías, Graciela, 2005. "Singular random matrix decompositions: Jacobians," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 296-312, April.
    18. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2016. "Unified improvements in estimation of a normal covariance matrix in high and low dimensions," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 233-248.
    19. Bodnar, Taras & Mazur, Stepan & Muhinyuza, Stanislas & Parolya, Nestor, 2017. "On the product of a singular Wishart matrix and a singular Gaussian vector in high dimensions," Working Papers 2017:7, Örebro University, School of Business.
    20. Taras Bodnar & Arjun Gupta, 2013. "An exact test for a column of the covariance matrix based on a single observation," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(6), pages 847-855, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:98:y:2007:i:8:p:1684-1692. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.