Seminar by Prof. Graczyk

This is a common work with H. Ishi and S. Mamane [1].

Graphical normal models were introduced by Lauritzen [2] and are one of the most important tools of contemporary statistics.  We give an outline of the modern theory of Wishart covariance estimators for graphical normal models, introduced in 2007 by Letac and Massam [3].  Our work grew up from the need to simplify, complete and extend the difficult paper of Letac and Massam [3], by applying harmonic analysis on cones.

Most of our results are for the graph G = An = 1-2-…-n.   These are fundamental classes of non-homogeneous cones appearing in the statistical theory of graphical models[2], corresponding to the practical model of nearest neighbor interactions. In the Gaussian character (X1, X2, . . . , Xn), non-neighbours Xi, Xj , |i − j| > 1 are conditionally independent with respect to other variables.

The methods of proof are simple.  They are based on inductive changes of variables and using natural ”future” and ”past” power functions δ(M) and ∆(M) on the cones that we consider.

[1] Graczyk, P., Ishi, H. and Mamane, S. (2016), Riesz and Wishart distributions on the cones related to An graphs, Preprint.
[2] Lauritzen, S.L. (1996), Graphical models, Oxford University Press.
[3] Letac, G. and Massam, H. (2007), Wishart distributions for decomposable graphs, The Annals of Statistics, vol. 35, no. 3, pp. 1278-1323.