Optimization of the size of nonsensitivness regions.

*(English)*Zbl 1091.62521The 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.

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) |

##### 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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