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# Large Sample Theory: Freedom-Equation Hypotheses

• George A. F. Seber
Part of the Springer Series in Statistics book series (SSS)

## 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 × pq matrix with (i, j)th element $$\partial \theta _{i}/\partial \alpha _{j}$$, which we assume to have rank pq. 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
$$\displaystyle{-n^{-1}E_{\boldsymbol{\theta }}\left [\frac{\partial ^{2}L(\boldsymbol{\theta })} {\partial \theta _{i}\partial \theta _{j}} \right ] = -E\left [\frac{\partial ^{2}\log \,f(x,\boldsymbol{\theta })} {\partial \theta _{i}\partial \theta _{j}} \right ],}$$
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
$$\displaystyle{\mathbf{B}_{\boldsymbol{\alpha }} = [\boldsymbol{\varTheta }'\mathbf{B}_{\boldsymbol{\theta }}\boldsymbol{\varTheta }]_{\alpha } =\boldsymbol{\varTheta } _{\boldsymbol{\alpha }}'\mathbf{B}_{\boldsymbol{\theta }(\boldsymbol{\alpha })}\boldsymbol{\varTheta }_{\boldsymbol{\alpha }}.}$$
We note that
$$\displaystyle{\mathbf{D}_{\alpha }L(\boldsymbol{\theta }) =\boldsymbol{\varTheta } _{\alpha }'\mathbf{D}_{\boldsymbol{\theta }}L(\boldsymbol{\theta }(\boldsymbol{\alpha })).}$$

## Reference

1. Seber, G. A. F. (1964). The linear hypothesis and large sample theory. Annals of Mathematical Statistics, 35(2), 773–779.

## Copyright information

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• George A. F. Seber
• 1
1. 1.Department of StatisticsThe University of AucklandAucklandNew Zealand