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Wishart exponential families on cones related to tridiagonal matrices

Author

Listed:
  • Piotr Graczyk

    (University of Angers)

  • Hideyuki Ishi

    (Nagoya University, Furo-cho)

  • Salha Mamane

    (University of the Witwatersrand)

Abstract

Let G be the graph corresponding to the graphical model of nearest neighbor interaction in a Gaussian character. We study Natural Exponential Families (NEF) of Wishart distributions on convex cones $$Q_G$$ Q G and $$P_G$$ P G , where $$P_G$$ P G is the cone of tridiagonal positive definite real symmetric matrices, and $$Q_G$$ Q G is the dual cone of $$P_G$$ P G . The Wishart NEF that we construct include Wishart distributions considered earlier for models based on decomposable(chordal) graphs. Our approach is, however, different and allows us to study the basic objects of Wishart NEF on the cones $$Q_G$$ Q G and $$P_G$$ P G . We determine Riesz measures generating Wishart exponential families on $$Q_G$$ Q G and $$P_G$$ P G , and we give the quadratic construction of these Riesz measures and exponential families. The mean, inverse-mean, covariance and variance functions, as well as moments of higher order, are studied and their explicit formulas are given.

Suggested Citation

  • Piotr Graczyk & Hideyuki Ishi & Salha Mamane, 2019. "Wishart exponential families on cones related to tridiagonal matrices," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(2), pages 439-471, April.
    • Handle: RePEc:spr:aistmt:v:71:y:2019:i:2:d:10.1007_s10463-018-0647-z
    DOI: 10.1007/s10463-018-0647-z
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    References listed on IDEAS

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    1. Tsukuma, Hisayuki & Konno, Yoshihiko, 2006. "On improved estimation of normal precision matrix and discriminant coefficients," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1477-1500, August.
    2. Konno, Yoshihiko, 2009. "Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2237-2253, November.
    3. Konno, Yoshihiko, 2007. "Estimation of normal covariance matrices parametrized by irreducible symmetric cones under Stein's loss," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 295-316, February.
    4. Nariaki Sugiura & Yoshihiko Konno, 1988. "Entropy loss and risk of improved estimators for the generalized variance and precision," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 40(2), pages 329-341, June.
    5. Satoshi Kuriki & Yasuhide Numata, 2010. "Graph presentations for moments of noncentral Wishart distributions and their applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(4), pages 645-672, August.
    6. Andersson, Steen A. & Klein, Thomas, 2010. "On Riesz and Wishart distributions associated with decomposable undirected graphs," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 789-810, April.
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