The usual likelihood computations for exponential family models, such as maximum likelihood estimators, standard errors, the deviance, residuals and influence, are all covered in this book. However the major theme of the book is the development of second order asymptotic inference in which nonlinearity is given a geometric interpretation. There are two major strands to the study of curvature and geometry in the statistical literature. This strand treats general probability families as points on abstract manifolds equipped with Reimannian metrics. This approach can handle quite general statistical models, not just exponential family models.
Different connections lead to essentially different definitions for curvature.
This richness does have some advantages. The choice of connection can be tuned to the property of the maximum likelihood estimator which is it is desired to study bias, skewness, normality etc. This book follows this second strand, taking advantage of strong analogies between exponential family nonlinear models and nonlinear regression. Wei identifies briefly the relationship between the two approaches. But is it far much more direct and uses a much smaller mathematical apparatus.