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Wishart and Pseudo-Wishart Distributions and Some Applications to Shape Theory

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  • Díaz-García, José A.
  • Jáimez, Ramón Gutierrez
  • Mardia, Kanti V.

Abstract

Suppose thatX~N-m([mu], [Sigma], [Theta]). An expression for the density function is given when[Sigma][greater-or-equal, slanted]0 and/or[Theta]:[greater-or-equal, slanted]0. An extension of Uhlig's result (Uhlig [17]) is expanded for the singular value decomposition of a matrixZof orderN-mwhen the rank (Z)=q[less-than-or-equals, slant]min(N, m). This paper fills an important gap in unifying, for the first time, all Wishart and pseudo-Wishart distributions, whether central or noncentral, whether singular or nonsingular, and applying them in shape analysis. In particular, the shape density and the size-and-shape cone density are obtained for the singular general case.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:63:y:1997:i:1:p:73-87
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    References listed on IDEAS

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    1. Khatri, C. G., 1979. "Characterizations of multivariate normality II. Through linear regressions," Journal of Multivariate Analysis, Elsevier, vol. 9(4), pages 589-598, December.
    2. Goodall, Colin & Mardia, Kanti V., 1992. "The noncentral Bartlett decompositions and shape densities," Journal of Multivariate Analysis, Elsevier, vol. 40(1), pages 94-108, January.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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).
    6. 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.
    7. Zhang, Zhihua, 2007. "Pseudo-inverse multivariate/matrix-variate distributions," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1684-1692, September.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. Hisayuki Tsukuma & Tatsuya Kubokawa, 2014. "Unified Improvements in Estimation of a Normal Covariance Matrix in High and Low Dimesions," CIRJE F-Series CIRJE-F-937, CIRJE, Faculty of Economics, University of Tokyo.
    15. 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.
    16. 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.
    17. Díaz-García, José A. & Gutiérrez Jáimez, Ramón, 2008. "Singular matrix variate beta distribution," Journal of Multivariate Analysis, Elsevier, vol. 99(4), pages 637-648, April.
    18. 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.
    19. 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.
    20. 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.
    21. 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.
    22. 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.
    23. 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.

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