# Inference Properties

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

## Abstract

We assume the model $$\mathbf{y} =\boldsymbol{\theta } +\boldsymbol{\varepsilon }$$, $$G:\boldsymbol{\theta }\in \varOmega$$, a p-dimensional vector space in $$\mathbb{R}^{n}$$, and $$H:\boldsymbol{\theta }\in \omega$$, a pq dimensional subspace of Ω; $$\boldsymbol{\varepsilon }$$ is $$N_{n}[\mathbf{0},\sigma ^{2}\mathbf{I}_{n}]$$. To test H we choose a region W called the critical region and we reject H if and only if y ∈ W. The power of the test $$\beta (W,\boldsymbol{\theta })$$ is defined to be probability of rejecting H when $$\boldsymbol{\theta }$$ is the true value of $$\mathrm{E}[\mathbf{y}]$$. Thus,
$$\displaystyle{\beta (W,\boldsymbol{\theta }) =\Pr [\mathbf{y} \in W\vert \boldsymbol{\theta }]}$$
and is a function of W and $$\boldsymbol{\theta }$$. The size of a critical region W is $$\sup _{\boldsymbol{\theta }\in W}\beta (W,\boldsymbol{\theta })$$, and if $$\beta (W,\boldsymbol{\theta }) =\alpha$$ for all $$\boldsymbol{\theta }\in \omega$$, then W is said to be a similar region of size α. If W is of size α and $$\beta (W,\boldsymbol{\theta }) \geq \alpha$$ for every $$\boldsymbol{\theta }\in \varOmega -\omega$$ (the set of all points in Ω which are not in ω), then W is said to be unbiased. In particular, if we have the strict inequality $$\beta (W,\boldsymbol{\theta }) >\alpha$$ for $$\boldsymbol{\theta }\in \varOmega -\omega$$, then W is said to be consistent. Finally we define W to be a uniformly most powerful (UMP) critical region of a given class C if W ∈ C and if, for any W′ ∈ C and all $$\boldsymbol{\theta }\in \varOmega -\omega$$,
$$\displaystyle{\beta (W,\boldsymbol{\theta }) \geq \beta (W',\boldsymbol{\theta }).}$$
Obviously a wide choice of W is possible for testing H, and so we would endeavor to choose a critical region which has some, or if possible, all of the desired properties mentioned above, namely similarity, unbiasedness or consistency, and providing a UMP test for certain reasonable classes of critical regions. Other criteria such as invariance are also used (Lehmann and Romano 2005). The F-test for H, given by
$$\displaystyle{F = \frac{f_{2}} {f_{1}} \frac{\mathbf{y}'(\mathbf{P}_{\varOmega } -\mathbf{P}_{\omega })\mathbf{y}} {\mathbf{y}'(\mathbf{I}_{n} -\mathbf{P}_{\varOmega })\mathbf{y}},}$$
where f1 = q and $$f_{2} = n - p$$, provides such a critical region W0, say, and we now consider some properties of W0.

## References

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