The Linear Model and Hypothesis pp 149-174 | Cite as

# Large Sample Theory: Constraint-Equation Hypotheses

## Abstract

Apart from Chap. 8 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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