5. Inference Properties

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, 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 \(\boldsymbol{\theta }\in \varOmega -\omega\), 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 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.