Linear regression.
type: Item
instance of: scholarly article
title: Linear regression.
zbMATH Open document ID: 1039.62061
published in: Lecture Notes in Statistics
publication date: 2003-08-13T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 62J20
Mathematics Subject Classification ID: 62-01
zbMATH DE Number: 1963169
zbMATH Keywords: risk
zbMATH Keywords: admissibility
zbMATH Keywords: squared error loss
zbMATH Keywords: coefficient of determination
zbMATH Keywords: predictive loss
zbMATH Keywords: normality
zbMATH Keywords: linear unbiased estimation
zbMATH Keywords: Gauss-Markov theorem
zbMATH Keywords: consistency
zbMATH Keywords: biased estimation
zbMATH Keywords: reparameterization
zbMATH Keywords: Stein estimation
zbMATH Keywords: pretest estimation
zbMATH Keywords: Marquardt estimator
zbMATH Keywords: jackknife modified ridge estimator
zbMATH Keywords: iteration estimator
zbMATH Keywords: linear minimax estimator
zbMATH Keywords: weighted least squares estimation
zbMATH Keywords: M estimator
zbMATH Keywords: linear admissibility
zbMATH Keywords: non-scalar covariance matrix
zbMATH Keywords: two-stage estimation
zbMATH Keywords: seemingly unrelated regression
zbMATH Keywords: heteroscedastic errors
zbMATH Keywords: equicorrelated errors
zbMATH Keywords: autocorrelated errors
zbMATH Keywords: singular covariance matrix
zbMATH Keywords: generalized least squares
zbMATH Keywords: regression diagnostics
zbMATH Keywords: Mallows' \(C_p\)
zbMATH Keywords: stepwise regression
zbMATH Keywords: goodness of fit
zbMATH Keywords: variance inflation factors
zbMATH Keywords: scaled condition indexes
zbMATH Keywords: normal quantile plot
zbMATH Keywords: residual plot
zbMATH Keywords: leverage
zbMATH Keywords: influential observations
zbMATH Keywords: testing model assumptions
author: Jürgen Gross
reviewed by: Yuehua Wu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1963169