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Smyth, G. K. (2000). Review of "Exponential Family Nonlinear Models" by Bo-Cheng Wei. Austral. J. Statist. 42, 500.
Exponential Family Nonlinear Models
By Bo-Cheng Wei. 1998.
x+230 pages. A$? (softcover).
Generalized linear models provide a very popular method for modelling non-normal data making maximal use of analogies with normal linear regression. What makes these models ``linear'' is use of a linear predictor for a suitable function of the expected values of the observations. If we abandon this assumption of link-linearity and instead allow the expected values to be general functions of the regression parameters then we arrive at generalized nonlinear models or, in the terminology of this book, exponential family nonlinear models. Although an obvious extension of generalized linear models, there have been so far only a handful of papers published on exponential family nonlinear models. Here we have a full book length treatment of these models.
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. The first started with Efron (1975) and Amari (1985). 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. However there is a disturbing indeterminancy to these models which arises from the lack of a unique way to define translation on manifolds (connections). 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.
Meanwhile Beale (1960) and Bates and Watts (BW) (1980) developed a direct approach to curvature for nonlinear least squares problems. 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. Wei's curvature in the mean-parameter space is essentially equivalent to the use of an α-connection in the Efron-Amari approach with α = -1, while Wei's curvature in the canonical parameter space is equivalent to an α-connection with α=1. The BW-Wei approach is much less general than that of Efron-Amari. Generalized nonlinear models are probably the most general class of models to which the BW-Wei approach can be applied, and it doesn't seem to extend to connections other than ± 1. However ± 1 seems to be enough for the purpose of deriving asymptotic expansions for maximum likelihood estimators. Wei's approach is also more ``messy'' than the differential geometric approach in that it involves lots of long matrix and 3-way array expressions. But is it far much more direct and uses a much smaller mathematical apparatus. Most statistical readers will be able to go faster and further with this approach that with the Reimannian manifold approach. The variety of results in this book for non iid regression models is evidence of this.
The first two, fairly short, chapters of the book define exponential family nonlinear models and derive the usual first order likelihood calculations. Chapter 3 introduces the geometric framework and defines intrinsic and parameter effects curvatures analogous to Bates and Watts (1980). Chapter 4 develops bias and variance adjustments for maximum likelihood estimators and approximate expressions for information loss due to curvature. Chapter 5 develops approximate confident regions analogous to those of Hamilton, Watts and Bates (1982) and Hamilton (1986) for normal models. Chapter 6 is a wide ranging chapter studying case deletion, local influence, generalized leverage and varying dispersion. Chapter 7 briefly discusses extensions of the approach to multinomial, quasi-likelihood and other models.
Despite the extensive results in this book there is still much work to be done in the area. For example, some of the approximations given in the later parts of the book do not come with specified accuracy (such as O(n-1/2)) and it is not clear in what circumstances they will be accurate enough for practical use.
The text was prepared camera-ready by the author. I found the book to be generally well written and easy to read, even given the copious matrix and array expressions. There are though some minor inaccuracies of expression, mainly grammatical but also sometimes mathematical. These can be distracting at times but won't hold up a reader for long.
As the author says in the introduction, this is a theoretical book. Although the book is enhanced by some simulation studies and data examples, these tend to be bit cursory. The data examples tend to illustrate the theoretical computations rather than being serious analyses of the data at hand. So the book is not in its current form a guide to practitioners. Nor is it designed to be a textbook. However as a research monograph it will be of considerable interest and can be recommended to researchers in generalized linear models, statistical curvature and second order inference.
University of Queensland