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Erschienen in:
2015 | OriginalPaper | Buchkapitel
11. Large Sample Theory: Freedom-Equation Hypotheses
verfasst von : George A. F. Seber
Erschienen in: The Linear Model and Hypothesis
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Abstract
In this chapter we assume once again that \(\boldsymbol{\theta }\in W\). However our hypothesis H now takes the form of freedom equations, namely \(\boldsymbol{\theta }=\boldsymbol{\theta } (\boldsymbol{\alpha })\), where \(\boldsymbol{\alpha }= (\alpha _{1},\alpha _{2},\ldots,\alpha _{p-q})'\). We require the following additional notation. Let \(\boldsymbol{\Theta }_{\boldsymbol{\alpha }}\) be the p × p − q matrix with (i, j)th element \(\partial \theta _{i}/\partial \alpha _{j}\), which we assume to have rank p − q. As before, \(L(\boldsymbol{\theta }) =\log \prod _{ i=1}^{n}f(x_{i},\boldsymbol{\theta })\) is the log likelihood function. Let \(\mathbf{D}_{\boldsymbol{\theta }}L(\boldsymbol{\theta })\) and \(\mathbf{D}_{\boldsymbol{\alpha }}L(\boldsymbol{\theta }(\boldsymbol{\alpha }))\) be the column vectors whose ith elements are \(\partial L(\boldsymbol{\theta })/\partial \theta _{i}\) and \(\partial L(\boldsymbol{\theta })/\partial \alpha _{i}\) respectively. As before, \(\mathbf{B}_{\boldsymbol{\theta }}\) is the p × p information matrix with i, jth element and we add \(\mathbf{B}_{\boldsymbol{\alpha }}\), the \(p - q \times p - q\) information matrix with i, jth element \(-E[\partial ^{2}\log \,f(x,\boldsymbol{\theta }(\boldsymbol{\alpha }))/\partial \alpha _{i}\partial \alpha _{j}]\). To simplify the notation we use \([\cdot ]_{\boldsymbol{\alpha }}\) to denote that the matrix in square brackets is evaluated at \(\boldsymbol{\alpha }\), for example We note that
$$\displaystyle{-n^{-1}E_{\boldsymbol{\theta }}\left [\frac{\partial ^{2}L(\boldsymbol{\theta })} {\partial \theta _{i}\partial \theta _{j}} \right ] = -E\left [\frac{\partial ^{2}\log \,f(x,\boldsymbol{\theta })} {\partial \theta _{i}\partial \theta _{j}} \right ],}$$
$$\displaystyle{\mathbf{B}_{\boldsymbol{\alpha }} = [\boldsymbol{\varTheta }'\mathbf{B}_{\boldsymbol{\theta }}\boldsymbol{\varTheta }]_{\alpha } =\boldsymbol{\varTheta } _{\boldsymbol{\alpha }}'\mathbf{B}_{\boldsymbol{\theta }(\boldsymbol{\alpha })}\boldsymbol{\varTheta }_{\boldsymbol{\alpha }}.}$$
$$\displaystyle{\mathbf{D}_{\alpha }L(\boldsymbol{\theta }) =\boldsymbol{\varTheta } _{\alpha }'\mathbf{D}_{\boldsymbol{\theta }}L(\boldsymbol{\theta }(\boldsymbol{\alpha })).}$$