Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras

• 成果类型: Article
• 署名作者: Massam, H; Neher, E
• 署名单位: York University - Canada; University of Virginia; University of Ottawa
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1998
• 页码: 1051-1082
• 关键词: symmetrical cones; distributions

摘要

In this paper we generalize the major results of Andersson and Perlman on LCI models to the setting of symmetric cones and give an explicit closed form formula for the estimate of the covariance matrix in the generalized LCI models that we define. To this end, we replace the cone H-I(+)(R) sitting inside the Jordan algebra of symmetric real I x I-matrices by the symmetric cone Omega of an Euclidean Jordan algebra V. We introduce the Andersson-Perlman cone Omega(K) subset of or equal to Omega which generalizes P(K) subset of or equal to H-I(+)(R). We prove several characterizations and properties of Omega(K) which allows us to recover, though with different proofs, the main results of Andersson and Perlman regarding P(K). The new lattice conditional independence models are defined, assuming that the Euclidean Jordan algebra V has a symmetric representation. Using standard results from the theory of Jordan algebras, we can reduce the general model to the case where V is the Jordan algebra of Hermitian matrices over the real, complex or quaternionic numbers, and Omega is the corresponding cone of positive-defiinite matrices. Our main statistical result is a closed-form formula for the estimate of the covariance matrix in the generalized LCI model. We also give the likelihood ratio test for testing a given model versus another one, nested within the first.