,Dale L. Zimmerman
Department of Statistics and Actuarial
Science
University of Iowa
Iowa City, IA, USA
ISBN 978-3-030-52073-1 ISBN 978-3-030-52074-8 (eBook)
https://doi.org/10.1007/978-3-030-52074-8
Mathematics Subject Classification: 62J05, 62J10, 62F03, 62F10, 62F25
© Springer Nature Switzerland AG 2020
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, expressed or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.
This Springer imprint is published by the registered company Springer Nature Switzerland AG.
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
,Contents
1 A Brief Introduction ....................................................................................... 1
2 Selected Matrix Algebra Topics and Results................................................. 3
3 Generalized Inverses and Solutions to Systems
of Linear Equations ........................................................................................ 7
4 Moments of a Random Vector and of Linear and Quadratic
Forms in a Random Vector .......................................................................... 21
5 Types of Linear Models ................................................................................ 31
6 Estimability ................................................................................................... 39
7 Least Squares Estimation for the Gauss–Markov Model ........................... 63
8 Least Squares Geometry and the Overall ANOVA .................................... 91
9 Least Squares Estimation and ANOVA for Partitioned Models .............. 103
10 Constrained Least Squares Estimation and ANOVA ............................... 131
11 Best Linear Unbiased Estimation for the Aitken Model .......................... 153
12 Model Misspecification ............................................................................... 171
13 Best Linear Unbiased Prediction ............................................................... 185
14 Distribution Theory..................................................................................... 223
15 Inference for Estimable and Predictable Functions ................................. 255
16 Inference for Variance–Covariance Parameters ....................................... 325
17 Empirical BLUE and BLUP ...................................................................... 351
vii
, A Brief Introduction
1
This book contains 296 solved exercises on the theory of linear models. The
exercises are taken from the author’s graduate-level textbook, Linear Model Theory:
With Examples and Exercises, which was published by Springer in 2020. The
exercises themselves have been restated, when necessary and feasible, to make them
as comprehensible as possible independently of the textbook, but the solutions refer
liberally to theorems and other results therein. They are arranged in chapters, the
numbers and titles of which are identical to those of the chapters in the textbook
that have exercises.
Some of the exercises and solutions are short, while others have multiple parts
and are quite lengthy. Some are proofs of theorems presented but not proved in the
aforementioned textbook, but most are specializations of said theorems and other
general results to specific linear models. In this respect they are quite similar to the
textbook’s examples. A few of the exercises require the use of a computer, but none
involve the analysis of actual data.
The author is not aware of any other published set of solved exercises for a
graduate-level course on the theory of linear models. It is hoped that students
and instructors alike, possibly even those not using Linear Model Theory: With
Examples and Exercises for their course, will find these exercises and solutions
useful.
© Springer Nature Switzerland AG 2020 1
D. L. Zimmerman, Linear Model Theory,
https://doi.org/10.1007/978-3-030-52074-8_1
Department of Statistics and Actuarial
Science
University of Iowa
Iowa City, IA, USA
ISBN 978-3-030-52073-1 ISBN 978-3-030-52074-8 (eBook)
https://doi.org/10.1007/978-3-030-52074-8
Mathematics Subject Classification: 62J05, 62J10, 62F03, 62F10, 62F25
© Springer Nature Switzerland AG 2020
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, expressed or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.
This Springer imprint is published by the registered company Springer Nature Switzerland AG.
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
,Contents
1 A Brief Introduction ....................................................................................... 1
2 Selected Matrix Algebra Topics and Results................................................. 3
3 Generalized Inverses and Solutions to Systems
of Linear Equations ........................................................................................ 7
4 Moments of a Random Vector and of Linear and Quadratic
Forms in a Random Vector .......................................................................... 21
5 Types of Linear Models ................................................................................ 31
6 Estimability ................................................................................................... 39
7 Least Squares Estimation for the Gauss–Markov Model ........................... 63
8 Least Squares Geometry and the Overall ANOVA .................................... 91
9 Least Squares Estimation and ANOVA for Partitioned Models .............. 103
10 Constrained Least Squares Estimation and ANOVA ............................... 131
11 Best Linear Unbiased Estimation for the Aitken Model .......................... 153
12 Model Misspecification ............................................................................... 171
13 Best Linear Unbiased Prediction ............................................................... 185
14 Distribution Theory..................................................................................... 223
15 Inference for Estimable and Predictable Functions ................................. 255
16 Inference for Variance–Covariance Parameters ....................................... 325
17 Empirical BLUE and BLUP ...................................................................... 351
vii
, A Brief Introduction
1
This book contains 296 solved exercises on the theory of linear models. The
exercises are taken from the author’s graduate-level textbook, Linear Model Theory:
With Examples and Exercises, which was published by Springer in 2020. The
exercises themselves have been restated, when necessary and feasible, to make them
as comprehensible as possible independently of the textbook, but the solutions refer
liberally to theorems and other results therein. They are arranged in chapters, the
numbers and titles of which are identical to those of the chapters in the textbook
that have exercises.
Some of the exercises and solutions are short, while others have multiple parts
and are quite lengthy. Some are proofs of theorems presented but not proved in the
aforementioned textbook, but most are specializations of said theorems and other
general results to specific linear models. In this respect they are quite similar to the
textbook’s examples. A few of the exercises require the use of a computer, but none
involve the analysis of actual data.
The author is not aware of any other published set of solved exercises for a
graduate-level course on the theory of linear models. It is hoped that students
and instructors alike, possibly even those not using Linear Model Theory: With
Examples and Exercises for their course, will find these exercises and solutions
useful.
© Springer Nature Switzerland AG 2020 1
D. L. Zimmerman, Linear Model Theory,
https://doi.org/10.1007/978-3-030-52074-8_1
