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Erschienen in:
2015 | OriginalPaper | Buchkapitel
3. Estimation
verfasst von : George A. F. Seber
Erschienen in: The Linear Model and Hypothesis
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Abstract
Suppose we have the model \(\mathbf{y} =\boldsymbol{\theta } +\boldsymbol{\varepsilon }\), where \(\mathrm{E}[\boldsymbol{\varepsilon }] = \mathbf{0}\), \(\mathrm{Var}[\boldsymbol{\varepsilon }] =\sigma ^{2}\mathbf{I}_{n}\), and \(\boldsymbol{\theta }\in \varOmega\), a p-dimensional vector space. One reasonable estimate of \(\boldsymbol{\theta }\) would be the value \(\hat{\boldsymbol{\theta }}\), called the least squares estimate, that minimizes the total “error” sum of squares subject to \(\boldsymbol{\theta }\in \varOmega\). A clue as to how we might calculate \(\hat{\boldsymbol{\theta }}\) is by considering the simple case in which y is a point P in three dimensions and Ω is a plane through the origin O. We have to find the point Q (\(=\hat{\boldsymbol{\theta }}\)) in the plane so that PQ
2 is a minimum; this is obviously the case when OQ is the orthogonal projection of OP onto the plane. This idea can now be generalized in the following theorem.
$$\displaystyle{SS =\sum _{ i=1}^{n}\varepsilon _{ i}^{2} =\parallel \mathbf{y}-\boldsymbol{\theta }\parallel ^{2}}$$