Plane Answers to Complex Questions pp 1-16 | Cite as

# Introduction

Chapter

First Online:

## Abstract

This book is about linear models. Linear models are models that are linear in their parameters. A typical model considered is where

$$ {Y}={X}\beta + {e},$$

*Y*is an*n*× 1 vector of random observations,*X*is an*n*×*p*matrix of known constants called the*model*(or*design*) matrix,*β*is a*p*× 1 vector of unobservable fixed parameters, and*e*is an*n*× 1 vector of unobservable random errors. Both*Y*and*e*are random vectors. We assume that the errors have mean zero, a common variance, and are uncorrelated. In particular, E(*e*) = 0 and Cov(*e*) = σ_{2}*I*, where σ_{2}is some unknown parameter. (The operations E(·) and Cov(·) will be defined formally a bit later.) Our object is to explore models that can be used to predict future observable events. Much of our effort will be devoted to drawing inferences, in the form of point estimates, tests, and confidence regions, about the parameters β and σ_{2}. In order to get tests and confidence regions, we will assume that*e*has an*n*-dimensional normal distribution with mean vector (0,0, …, 0)’ and covariance matrix σ_{2}*I*, i.e.,*e*~*N*(0, σ_{2}*I*).## Keywords

Covariance Matrix Generalize Linear Model Quadratic Form Random Vector Multivariate Normal Distribution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media, LLC 2011