Differential geometry provides an aesthetically appealing and oftenrevealing view of statistical inference. Beginning with anelementary treatment of one-parameter statistical models and endingwith an overview of recent developments, this is the first book toprovide an introduction to the subject that is largely accessibleto readers not already familiar with differential geometry. It alsogives a streamlined entry into the field to readers with richermathematical backgrounds. Much space is devoted to curvedexponential families, which are of interest not only because theymay be studied geometrically but also because they are analyticallyconvenient, so that results may be derived rigorously. In addition,several appendices provide useful mathematical material on basicconcepts in differential geometry. Topics covered include thefollowing:
Basic properties of curved exponential families
Elements of second-order, asymptotic theory
The Fisher-Efron-Amari theory of information loss and recovery
Jeffreys-Rao information-metric Riemannian geometry
Curvature measures of nonlinearity
Geometrically motivated diagnostics for exponential familyregression
Geometrical theory of divergence functions
A classification of and introduction to additional work in thefield
Basic properties of curved exponential families
Elements of second-order, asymptotic theory
The Fisher-Efron-Amari theory of information loss and recovery
Jeffreys-Rao information-metric Riemannian geometry
Curvature measures of nonlinearity
Geometrically motivated diagnostics for exponential familyregression
Geometrical theory of divergence functions
A classification of and introduction to additional work in thefield
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Overview and Preliminaries.
ONE-PARAMETER CURVED EXPONENTIAL FAMILIES.
First-Order Asymptotics.
Second-Order Asymptotics.
MULTIPARAMETER CURVED EXPONENTIAL FAMILIES.
Extensions of Results from the One-Parameter Case.
Exponential Family Regression and Diagnostics.
Curvature in Exponential Family Regression.
DIFFERENTIAL-GEOMETRIC METHODS.
Information-Metric Riemannian Geometry.
Statistical Manifolds.
Divergence Functions.
Recent Developments.
Appendices.
References.
Indexes.
ONE-PARAMETER CURVED EXPONENTIAL FAMILIES.
First-Order Asymptotics.
Second-Order Asymptotics.
MULTIPARAMETER CURVED EXPONENTIAL FAMILIES.
Extensions of Results from the One-Parameter Case.
Exponential Family Regression and Diagnostics.
Curvature in Exponential Family Regression.
DIFFERENTIAL-GEOMETRIC METHODS.
Information-Metric Riemannian Geometry.
Statistical Manifolds.
Divergence Functions.
Recent Developments.
Appendices.
References.
Indexes.
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Robert E. Kass Carnegie Mellon University.
Paul W. Vos School of Allied Health Sciences, East Carolina University.
Paul W. Vos School of Allied Health Sciences, East Carolina University.
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