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Erschienen in:
2014 | OriginalPaper | Buchkapitel
5. Akaike’s Information Criterion (AIC) and the Third Variance
verfasst von : Kunio Takezawa
Erschienen in: Learning Regression Analysis by Simulation
Verlag: Springer Japan
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Abstract
One statistic in selecting a multiple regression equation is Mallows’ C
p
, which is an approximation of the error variance of the estimates given by a multiple regression equation. This error variance multiplied by n is written as where H is a hat matrix (Eq. (4.13) (page 164)) and \(\tilde{\mathbf{y}}\) are true values, that is, \(\tilde{y}_{i} = E[y_{i}]\) holds. The relation H
2 = H (Eq. (4.28) (page 167)) is used here, along with Eq. (4.34) (page 174). In deriving Eq. (4.32) (page 174), \(\mathbf{H}\tilde{\mathbf{y}} =\tilde{ \mathbf{y}}\) was assumed. However, this relationship is not assumed here.
$$\displaystyle\begin{array}{rcl} E{\Bigl [\sum _{i=1}^{n}{(\hat{y}_{ i} - E[y_{i}])}^{2}\Bigr ]}& =& E{\Bigl [{\bigl (\mathbf{H}(\tilde{\mathbf{y}}+\boldsymbol{\epsilon }) -\tilde{{\mathbf{y}}\bigr )}}^{t}{\bigl (\mathbf{H}(\tilde{\mathbf{y}}+\boldsymbol{\epsilon }) -\tilde{\mathbf{y}}\bigr )}\Bigr ]} \\ & =& E{\Bigl [{\bigl ((\mathbf{H} -\mathbf{I})\tilde{\mathbf{y}} +{ \mathbf{H}\boldsymbol{\epsilon }\bigr )}}^{t}{\bigl ((\mathbf{H} -\mathbf{I})\tilde{\mathbf{y}} + \mathbf{H}\boldsymbol{\epsilon }\bigr )}\Bigr ]} \\ & =& \tilde{{\mathbf{y}}}^{t}(\mathbf{I} -\mathbf{H})\tilde{\mathbf{y}} + E{[\boldsymbol{\epsilon }}^{t}\mathbf{H}\boldsymbol{\epsilon }] \\ & =& \tilde{{\mathbf{y}}}^{t}(\mathbf{I} -\mathbf{H})\tilde{\mathbf{y}} + {(q + 1)\sigma }^{2}, {}\end{array}$$