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.