Fundamental equations for statistical submanifolds with applications to the Bartlett correction
type: Item
instance of: scholarly article
title: Fundamental equations for statistical submanifolds with applications to the Bartlett correction
zbMATH Open document ID: 0705.62014
DOI: 10.1007/BF00050660
published in: Annals of the Institute of Statistical Mathematics
publication date: 1989-01-01T00:00:00Z
review text: In the differential geometry literature, there is a set of equations that describe relationships between invariant quantities on the submanifolds and supermanifolds when Riemannian connections are used. The author extends these equations to statistical manifolds, manifolds on which a pair of dual connections is defined. \textit{P. McCullagh} and \textit{D. R. Cox} [Ann. Stat. 14, 1419-1430 (1986; Zbl 0615.62041)] have given geometric interpretations for the Bartlett correction in normal theory regression. In this paper the result is generalized to exponential family regression. Further the author extends this geometric interpretation to statistical manifolds outside of the exponential family. He used a local approximation of the statistical manifold by an exponential family which has locally the same first and second derivatives of the log likelihood function.
Mathematics Subject Classification ID: 62A01
Mathematics Subject Classification ID: 62J99
Mathematics Subject Classification ID: 53B21
Mathematics Subject Classification ID: 53B15
Mathematics Subject Classification ID: 53B05
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 4157670
zbMATH Keywords: dual geometries
zbMATH Keywords: curvatures
zbMATH Keywords: imbedding curvature tensor
zbMATH Keywords: submanifolds
zbMATH Keywords: supermanifolds
zbMATH Keywords: statistical manifolds
zbMATH Keywords: dual connections
zbMATH Keywords: Bartlett correction
zbMATH Keywords: normal theory regression
zbMATH Keywords: exponential family regression
zbMATH Keywords: local approximation
zbMATH Keywords: derivatives of the log likelihood function
reviewed by: Silvelyn Zwanzig
MaRDI profile type: Publication
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author: Paul Vos
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description: scientific article; zbMATH DE number 4157670