This book explains why geometry should enter into parametric statistics and how the theory of asymptotic expansions involves a form of higher-order differential geometry. It gives some new explanations of geometric ideas from a first principles point of view as far as geometry is concerned.
"This book does an excellent job of explaining... geometrical ideas from a statistical point of view, using statistical examples as opposed to, for example, dynamical examples to illustrate the geometric concepts. It should prove most helpful in enlarging the typical statistician's working geometric vocabulary."
-Short Book Reviews of the ISI
"It is particularly good on the... problem of when to use coordinates and indices, and when to use the more geometric coordinate-free methods. Examples of this are its treatments of the affine connection and of the string theory of Barndorff-Nielsen, both of which are excellent. The first of these concepts is probably the most sophisticated that has been used in statistical applications. It can be viewed in many different ways, not all of which are necessarily useful to the statistician; the approach taken in the book is both relevant and clear."
-Bulletin of London Mathematical Society