"Geometrical Foundations of Asymptotic Inference" by Robert E. Kass, Paul W. Vos

This book is dedicated to differential geometry and 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.

The book 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

Brief Contents
Preface
1 Overview and Preliminaries
PART I: ONE-PARAMETER CURVED EXPONENTIAL FAMILIES
2 First-Order Asymptotics
3 Second-Order Asymptotics
PART II: MULTFARAMETER CURVED EXPONENTIAL FAMILIES
4 Extensions of Results from the One-Parameter Case
5 Exponential Family Regression and Diagnostics
6 Curvature in Exponential Family Regression
PART III: DIFFERENTIAL-GEOMETRIC METHODS
7 Information-Metric Riemannian Geometry
8 Statistical Manifolds
9 Divergence Functions
10 Recent Developments
Appendix A: Diffeomorphisms and the Inverse Function Theorem
Appendix B: Arclength and Curvature of Curves
Appendix C: Basic Concepts in Differential Geometry
Appendix D: A Coordinate-Free Definition of Weak Sphericity
References
Symbol Index
Index