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)


62J05 Linear regression; mixed models
62H15 Hypothesis testing in multivariate analysis
62J10 Analysis of variance and covariance (ANOVA)


Zbl 0870.62056
Full Text: DOI EuDML


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