Geometrical Foundations of Asymptotic Inference

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 thefieldDie asymptotische Interferenz ist ein wichtiges statistisches Problem. Eine geometrische Formulierung ausgehend von einem einfache, präzisen und konzeptuellen Ansatz stellen die Autoren dieses Buches vor. 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. Aus dem Inhalt:
Overview and Preliminaries;
ONE-PARAMETER CURVED EXPONENTIAL FAMILIES: First-Order Asymptotics;
Second-Order Asymptotics;
MULTIPARAMETER CURVED EXPONENTIAL FAMILES: 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;
A -
Diffeomorphisms and The Inverse Function Theorem;
B -
Arclength and Curvature of Curves;
C -
Basic Concepts in Differential Geometry;
D -
A Coordinate-free Definition of Weak Sphericity. "I highly recommend this book to anyone interested in asymptoticinferences." (Statistics & Decisions, Vol.19 No. 3, 2001) ROBERT E. KASS is Professor and Head of the Department of Statistics at Carnegie Mellon University. PAUL W. VOS is Associate Professor of Biostatistics at East Carolina University. Both authors received their PhDs from the University of Chicago. Differential geometry provides an aesthetically appealing and often revealing view of statistical inference. Beginning with an elementary treatment of one-parameter statistical models and ending with an overview of recent developments, this is the first book to provide an introduction to the subject that is largely accessible to readers not already familiar with differential geometry. It also gives a streamlined entry into the field to readers with richer mathematical backgrounds. Much space is devoted to curved exponential families, which are of interest not only because they may be studied geometrically but also because they are analytically convenient, so that results may be derived rigorously. In addition, several appendices provide useful mathematical material on basic concepts in differential geometry. Topics covered include the following:

• 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 family regression
• Geometrical theory of divergence functions
• A classification of and introduction to additional work in the field