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Nonlinear statistical models. (English) Zbl 0808.62058
Mathematics and its Applications (Dordrecht). 254. Dordrecht: Kluwer Academic Publishers. Bratislava: Ister Science Press Ltd. ix, 259 p. (1993).
The subject of nonlinear regression can be considered to be a natural generalization of the linear model. Nonlinear regression models are used either because the observed variables depend nonlinearly on the input variables based on the evidence of a possible relation or because nonlinear modelling might give a better fit involving a lesser number of unknown parameters. There are many examples of nonlinear models in physics, chemistry, economics etc. Even though computations and statistical reasoning are more complicated in nonlinear models than in the linear models, nonlinear models are widely used due to the advent of computers and availability of statistical packages which make extensive computations feasible. The problem is to study the properties of estimators of unknown parameters. The book of A. R. Gallant, Nonlinear statistical models. (1987; Zbl 0611.62071), follows the traditional approach for the study of properties of estimators of parameters via Taylor expansions. An alternative approach via weak convergence of least squares processes was given by the reviewer in his book, ‘Asymptotic theory of statistical inference.’ (1987; Zbl 0604.62025).
Here, the author studies the problem making global use of differential- geometric methods. The book is based mostly on the research work of the author. The contents are as follows:
1. Linear regression models; 2. Linear methods in nonlinear regression models; 3. Univariate regression models; 4. The structure of a multivariate nonlinear regression model and properties of $$L_ 2$$- estimators; 5. Nonlinear regression models: computations of estimators and curvature; 6. Local approximations of probability densities and moments of estimators; 7. Global approximations of densities of $$L_ 2$$- estimators; 8. Statistical consequences of global approximations especially in flat models; 9. Nonlinear exponential families.
The book needs a basic knowledge of differential geometry and it is a welcome addition to the literature on the theory of nonlinear regression.

MSC:
 62J02 General nonlinear regression 62-02 Research exposition (monographs, survey articles) pertaining to statistics 53B99 Local differential geometry