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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

  1. Atiqullah, M. (1962). The estimation of residual variance in quadratically balanced least squares problems and the robustness of the F test. Biometrika, 49, 83–91.zbMATHMathSciNetCrossRefGoogle Scholar
  2. Lehmann, E. L., & Romano, J. P. (2005). Testing statistical hypotheses (3rd ed.). New York: Springer.zbMATHGoogle Scholar
  3. Saw, J. G. (1964). Some notes on variance-ratio tests of the general linear hypothesis. Biometrika, 51, 511–518.MathSciNetCrossRefGoogle Scholar
  4. Scheffé, H. (1959). The analysis of variance. New York: Wiley.zbMATHGoogle Scholar
  5. Seber, G. A. F., & Lee, A. J. (2003). Linear regression analysis (2nd ed.). New York: Wiley.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • George A. F. Seber
    • 1
  1. 1.Department of StatisticsThe University of AucklandAucklandNew Zealand

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