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Optimization of the size of nonsensitivness regions. (English) Zbl 1091.62521
The standard statistical procedures based on a linear regression model are influenced by the inaccuracy $$\delta \mathbf \vartheta$$ in the value of variance components. In the paper, its effect is considered on the risk of a standard test. Let the risk $$\alpha$$ be worse by $$\epsilon$$, i.e., let the level of the test be $$\alpha +\epsilon$$. The nonsensitivness region $$\mathcal R_\epsilon$$ is introduced by the condition that the risk does not exceed $$\alpha +\epsilon$$ for all $$\delta \mathbf \vartheta \in \mathcal R_\epsilon ,$$ see L. Kubáček [Appl. Math., Praha 41, 433–445 (1996; Zbl 0870.62056)] for details.
First, two lemmas concerning the distribution of quadratic forms are proved and used in what follows. Then a test concerning the value of a vectorial first order parameter $$\beta$$ of a normally distributed $$n$$-dimensional random vector $$Y \sim N_n(X\beta , \Sigma (\mathbf \vartheta ))$$ is considered (the design matrix $$X$$ is known). The aim of the paper is to optimize the size of the corresponding region $$\mathcal R_\epsilon$$. Two numerical examples complete the paper.
Reviewer: Ivan Saxl (Praha)

##### MSC:
 62J05 Linear regression; mixed models 62H15 Hypothesis testing in multivariate analysis 62J10 Analysis of variance and covariance (ANOVA)
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##### References:
 [1] J. P. Imhof: Computing the distribution of quadratic forms in normal variables. Biometrika 48 (1961), 419-426. · Zbl 0136.41103 [2] J. Janko: Statistical Tables. Academia, Praha, 1958. [3] L. Kubáček: Linear model with inaccurate variance components. Appl. Math. 41 (1996), 433-445. · Zbl 0870.62056 [4] C. R. Rao: Statistical Inference and Its Applications. J. Wiley, New York-London-Sydney, 1965. · Zbl 0137.36203
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