### Abstract

Let y, ∼ N(Bx_{i}, ∑), i = 1, 2,..., N, and y ∼ N(Bθ, ∑) be independent multivariate observations, where the x_{i}'s are known vectors, B, θ and ∑ are unknown, ∑ being a positive definite matrix. The calibration problem deals with statistical inference concerning θ and the problem that we have addressed is the construction of confidence regions. In this article, we have constructed a region for θ based on a criterion similar to that satisfied by a tolerance region. The situation where θ is possibly a nonlinear function, say h(ξ), of fewer unknown parameters denoted by the vector ξ, is also considered. The problem addressed in this context is the construction of a region for ξ. The numerical computations required for the practical implementation of our region are explained in detail and illustrated using an example. Limited numerical results indicate that our regions satisfy the coverage probability requirements of multiple-use confidence regions.

Original language | English |
---|---|

Pages (from-to) | 1989-2013 |

Number of pages | 25 |

Journal | Annals of Statistics |

Volume | 26 |

Issue number | 5 |

Publication status | Published - Oct 1998 |

Externally published | Yes |

### Fingerprint

### Keywords

- Calibration
- Matrix variate beta distribution
- Matrix variate F distribution
- Multiple-use confidence region
- Multivariate linear model
- Noncentral chi-square
- Tolerance region
- Wishart distribution

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Statistics*,

*26*(5), 1989-2013.

**Tolerance regions and multiple-use confidence regions in multivariate calibration.** / Mathew, Thomas; Sharma, Manoj Kumar; Nordström, Kenneth.

Research output: Contribution to journal › Article

*Annals of Statistics*, vol. 26, no. 5, pp. 1989-2013.

}

TY - JOUR

T1 - Tolerance regions and multiple-use confidence regions in multivariate calibration

AU - Mathew, Thomas

AU - Sharma, Manoj Kumar

AU - Nordström, Kenneth

PY - 1998/10

Y1 - 1998/10

N2 - Let y, ∼ N(Bxi, ∑), i = 1, 2,..., N, and y ∼ N(Bθ, ∑) be independent multivariate observations, where the xi's are known vectors, B, θ and ∑ are unknown, ∑ being a positive definite matrix. The calibration problem deals with statistical inference concerning θ and the problem that we have addressed is the construction of confidence regions. In this article, we have constructed a region for θ based on a criterion similar to that satisfied by a tolerance region. The situation where θ is possibly a nonlinear function, say h(ξ), of fewer unknown parameters denoted by the vector ξ, is also considered. The problem addressed in this context is the construction of a region for ξ. The numerical computations required for the practical implementation of our region are explained in detail and illustrated using an example. Limited numerical results indicate that our regions satisfy the coverage probability requirements of multiple-use confidence regions.

AB - Let y, ∼ N(Bxi, ∑), i = 1, 2,..., N, and y ∼ N(Bθ, ∑) be independent multivariate observations, where the xi's are known vectors, B, θ and ∑ are unknown, ∑ being a positive definite matrix. The calibration problem deals with statistical inference concerning θ and the problem that we have addressed is the construction of confidence regions. In this article, we have constructed a region for θ based on a criterion similar to that satisfied by a tolerance region. The situation where θ is possibly a nonlinear function, say h(ξ), of fewer unknown parameters denoted by the vector ξ, is also considered. The problem addressed in this context is the construction of a region for ξ. The numerical computations required for the practical implementation of our region are explained in detail and illustrated using an example. Limited numerical results indicate that our regions satisfy the coverage probability requirements of multiple-use confidence regions.

KW - Calibration

KW - Matrix variate beta distribution

KW - Matrix variate F distribution

KW - Multiple-use confidence region

KW - Multivariate linear model

KW - Noncentral chi-square

KW - Tolerance region

KW - Wishart distribution

UR - http://www.scopus.com/inward/record.url?scp=0032275568&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032275568&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032275568

VL - 26

SP - 1989

EP - 2013

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

IS - 5

ER -