Linear regression.
The book covers the basic theory of linear regression models and presents a comprehensive survey of different estimation techniques as alternatives and complements to least squares estimation. The relationship between different estimators is clearly described and categories of estimators are worked out in detail. Proofs are given for the most relevant results, and the presented methods are illustrated with the help of numerical examples and graphics. Special emphasis is laid on the practicability, and possible applications are discussed. In Chapter 1, a brief introduction to the theory of linear regression models is given and a small numerical example is provided. In addition, an introduction to the basics of decision theory is delivered. In Chapter 2, point estimation of regression parameters and common variance of random errors are studied, and special attention is given to the ordinary least squares estimator and the least squares variance estimator. Properties of the least squares estimator of the regression parameter vector are discussed in detail. The next two chapters consist of the second part of the book, which is devoted to alternatives to least squares estimation. In Chapter 3, a number of various estimators is introduced, such as restricted least squares estimator, principal components estimator, ridge estimator, shrinkage estimator, general ridge estimator, linear minimax estimator, linear Bayes estimator, and robust estimators. Besides, the properties of these estimators are compared with those of the ordinary least squares estimator. How to use one or more nonlinear variants of the estimator in question in practice is also demonstrated. Chapter 4 investigates the structure of linearly admissible estimators of regression parameters. Chapter 5 discusses the estimation of the common variance of the random errors in detail. Some relaxation of the assumptions on the covariance matrix of the error vector is also presented. Chapter 6 gives a short review of some regression diagnostic procedures, which includes topics on variable selection, goodness of fit, diagnosing collinearity, inspecting residuals, finding influential observations, and testing model assumptions. A review on matrix algebra and stochastic vectors plus an example analysis with R are given in the appendix. Each chapter contains exercises. This book could serve as a textbook for a course on linear models at graduate level. It is also an excellent reference book for professional, scientists, and researches in theoretical and applied statistics and applied mathematics.
scientific article; zbMATH DE number 1963169