Wishart exponential families on cones related to tridiagonal matrices

Piotr Graczyk (), Hideyuki Ishi () and Salha Mamane ()
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Piotr Graczyk: University of Angers
Hideyuki Ishi: Nagoya University, Furo-cho
Salha Mamane: University of the Witwatersrand

Annals of the Institute of Statistical Mathematics, 2019, vol. 71, issue 2, No 8, 439-471

Abstract: 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.

Keywords: Wishart distribution; Graphical model; Nearest neighbor interaction (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10463-018-0647-z

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