zbMATH — the first resource for mathematics

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)
PDF BibTeX Cite
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.