Transformation and weighting in regression. (English) Zbl 0666.62062

Monographs on Statistics and Applied Probability. New York etc.: Chapman and Hall. x, 249 p. (1988).
Linear and nonlinear regression models are among the most used models for analyzing data. Experience from several applications indicates that the assumptions made about the error terms are often violated. This monograph emphazises problems caused by a heterogeneous error variance and/or nonnormally distributed error terms. Methods discussed for analyzing data in these cases fall into two categories: methods based on weighting and methods based on transformation.
The book mainly consists of three parts. After an introductory chapter, methods based on weighting are considered. Here, the error variance is assumed to be a parametric function, here called the variance function, of some variables. One of the more important examples of this is when the variance is a function of the mean of the response variable. Chapter 2 concerns estimation of the regression parameters when parameters in the variance function are known, while Chapter 3 extends to the case when also the variance function parameters need to be estimated.
The second part of the book concerns the second approach to analyze heterogeneous data, viz. transformation. Here, the interesting transforming-both-sides approach is taken, in which the response variable as well as the models’ systematic part are transformed by the same transformation. The motivation for this approach is that the relationship between the response and the predictors is preserved, which is not the case when only one variable or one side of the regression equation is transformed. This is of particular importance when the model is treated as a working hypothesis. Chapter 4 explores the transformation-both-sides method in detail while Chapter 5 combines the ideas of weighting and transformation.
The weighting and transformation methods for analyzing heterogeneous data are found to be highly sensitive to unusual observations. This motivates a rather comprehensive study of influence diagnostics and robust estimation, which is the content of Chapter 6, the third part of the book. The book ends with some basic background on estimation and testing, which is sketched in Chapter 7, and last, but not the least of importance, a chapter indicating some open problems and directions for further research.
In conclusion, the monograph provides a careful review of the current literature on an important subject. The approaches for attaching the problems considered are all illustrated in examples using real data. The book is therefore an important contribution for all practical working statisticians.
Reviewer: H.Nyquist


62J02 General nonlinear regression
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62J05 Linear regression; mixed models