Generalized linear models. (English) Zbl 0588.62104

McCullagh, P.; Nelder, J. A.

Monographs on Statistics and Applied Probability. London-New York: Chapman and Hall. XIII, 261 p. (1983).

This book gives a complete coverage of generalized linear models. In part 1 and 2 the generalized linear models are introduced and the basic concepts are defined: linear prediction, link function, scale parameter, probability distribution, weights and general properties of the deviance as well as some elements for the analysis of the residuals.
Part 3 is devoted to Gaussian linear models considered in the case when the error variance is independent from the mean. Some particular aspects such as aliasing of the parameters are considered and the numerical aspects of the estimation are clearly studied.
In parts 4,5,6 and 7, classical generalized linear models are considered: binary data, polytomous data, log linear models and models with the gamma distribution. For each case the log likelihood functions are exhibited, the asymptotic results are recalled and the classical link functions are listed and discussed.
Properties of the quasi-likelihood function are studied in part 8. In part 9 some models for survival data are presented and in part 10 some models with non-linear parameters are considered. The last part is devoted to model checking.
In the whole book, the technical aspects take an important place and many numerical examples are carefully analysed and interpreted. Furthermore it is self contained since useful theoretical results are briefly recalled when needed.

Reviewer: J.R.Mathieu

MSC:

62Jxx	Linear inference, regression
62-02	Research exposition (monographs, survey articles) pertaining to statistics
62J99	Linear inference, regression
62F10	Point estimation
62-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62F03	Parametric hypothesis testing
62F05	Asymptotic properties of parametric tests
62F12	Asymptotic properties of parametric estimators

Keywords:

generalized linear models; linear prediction; link function; weights; residuals; Gaussian linear models; aliasing; binary data; polytomous data; log linear models; gamma distribution; log likelihood functions; asymptotic results; quasi-likelihood function; survival data; non-linear parameters; model checking; numerical examples