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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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