• Ronald Christensen
Part of the Springer Texts in Statistics book series (STS)


In this chapter, properties of least squares estimates are examined for the model
$$\begin{array}{*{20}{c}} {Y = X\beta + e,}&{E(e) = 0,}&{Cov(e)} \end{array} = {\sigma ^2}I$$
The chapter begins with a discussion of the concept of estimability in linear models. Section 2 characterizes least squares estimates. Sections 3, 4, and 5 establish that least squares estimates are best linear unbiased estimates, maximum likelihood estimates, and minimum variance unbiased estimates. The last two of these properties require the additional assumption e ~ N(0, σ 2 I)Section 6 also assumes that the errors are normally distributed and presents the distributions of various estimates. From these distributions various tests and confidence intervals are easily obtained. Section 7 examines the model
$$\begin{array}{*{20}{c}} {Y = X\beta + e,}&{E(e) = 0,}&{Cov(e)} \end{array} = {\sigma ^2}V$$
where V is a known positive definite matrix. Section 7 introduces weighted least squares estimates and presents properties of those estimates. Section 8 presents the normal equations and establishes their relationship to least squares and weighted least squares estimation. Section 9 discusses Bayesian estimation.


Mean Square Error Bayesian Analysis Unbiased Estimate Normal Equation Prediction Interval 


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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Ronald Christensen
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerqueUSA

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