ABSTRACT
Models of the form of the general linear model, and in particular those of the form of the Gauss–Markov or Aitken model, are often used to obtain point estimates of the unobservable quantities represented by various parametric functions. In many cases, the parametric functions are ones that are expressible in the form λ ′ β $ \lambda ^{\prime }\beta $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351264686/dcbe6876-28fa-432e-bea2-b6aed193917a/content/inline-math5_1.tif"/> , where λ = ( λ 1 , λ 2 , … , λ P ) ′ $ \lambda = (\lambda _{1}, \lambda _{2}, \ldots , \lambda _{P})^{\prime } $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351264686/dcbe6876-28fa-432e-bea2-b6aed193917a/content/inline-math5_2.tif"/> is a P-dimensional column vector of constants, or equivalently ones that are expressible in the form ∑ j = 1 P λ j β j $ \sum _{j=1}^{P} \lambda _{j}\beta _{j} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351264686/dcbe6876-28fa-432e-bea2-b6aed193917a/content/inline-math5_3.tif"/> . Models of the form of the G–M, Aitken, or general linear model may also be used to obtain predictions for future quantities; these would be future quantities that are represented by unobservable random variables with expected values of the form λ ′ β $ \lambda ^{\prime }\beta $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351264686/dcbe6876-28fa-432e-bea2-b6aed193917a/content/inline-math5_4.tif"/> . The emphasis in this chapter is on the G–M model (in which the only parameter other than β 1 , β 2 , … , β P $ \beta _{1}, \beta _{2}, \ldots , \beta _{P} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351264686/dcbe6876-28fa-432e-bea2-b6aed193917a/content/inline-math5_5.tif"/> is the standard deviation σ $ \sigma $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351264686/dcbe6876-28fa-432e-bea2-b6aed193917a/content/inline-math5_6.tif"/> ) and on what might be regarded as a classical approach to estimation and prediction.
