# Large Sample Theory: Constraint-Equation Hypotheses

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

## Abstract

Apart from Chap.  on nonlinear models we have been considering linear models and hypotheses. We now wish to extend those ideas to non-linear hypotheses based on samples of n independent observations $$x_{1},x_{2},\ldots,x_{n}$$ (these may be vectors) from a general probability density function $$f(x,\boldsymbol{\theta })$$, where $$\boldsymbol{\theta }= (\theta _{1},\theta _{2},\ldots,\theta _{p})'$$ and $$\boldsymbol{\theta }$$ is known to belong to W a subset of $$\mathbb{R}^{p}$$. We wish to test the null hypothesis H that $$\boldsymbol{\theta }_{T}$$, the true value of $$\boldsymbol{\theta }$$, belongs to W H , a subset of W, given that n is large. We saw in previous chapters that there are two ways of specifying H; either in the form of “constraint” equations such as $$\mathbf{a}(\boldsymbol{\theta }) = (a_{1}(\boldsymbol{\theta }),a_{2}(\boldsymbol{\theta }),\ldots,a_{q}(\boldsymbol{\theta }))' = \mathbf{0}$$, or in the form of “freedom” equations $$\boldsymbol{\theta }=\boldsymbol{\theta } (\boldsymbol{\alpha })$$, where $$\boldsymbol{\alpha }= (\alpha _{1},\alpha _{2},\ldots,\alpha _{p-q})'$$, or perhaps by a combination of both constraint and freedom equations. Although to any freedom-equation specification there will correspond a constraint-equation specification and vice versa, this relationship is often difficult to derive in practice, and therefore the two forms shall be dealt with separately in this and the next chapter.

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