Linear Model Theory: Exercises and Solutions (1st Edition, 2020 – Dale L. Zimmerman) | Complete E-Book PDF Download

,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

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,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