Inference Properties

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 p − q 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 f ₁ = q and $f_{2} = n - p$, provides such a critical region W ₀, say, and we now consider some properties of W ₀.