The Annals of Statistics

Tolerance regions and multiple-use confidence regions in multivariate calibration

Abstract

Let $\mathrm{y}~ N(Bx_i,\Sigma),i=1,2\ldots,N$, and $\mathrm{y}~N(B\theta, \Sigma)$ be independent multivariate observations, where the $x_i$'s are known vectors, $B,\theta$ and $\Sigma$ are unknown, $\Sigma$ being a positive definite matrix. The calibration problem deals with statistical inference concerning $\theta$ and the problem that we have addressed is the construction of confidence regions. In this article, we have constructed a region for $\theta$ based on a criterion similar to that satisfied by a tolerance region. The situation where $\theta$ is possibly a nonlinear function, say $\mathrm{h}(\xi)$ of fewer unknown parameters denoted by the vector $(\xi)$, is also considered. The problem addressed in this context is the construction of a region for $\xi$. 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 multiple­use confidence regions.

Article information

Source
Ann. Statist., Volume 26, Number 5 (1998), 1989-2013.

Dates
First available in Project Euclid: 21 June 2002

https://projecteuclid.org/euclid.aos/1024691366

Digital Object Identifier
doi:10.1214/aos/1024691366

Mathematical Reviews number (MathSciNet)
MR1673287

Zentralblatt MATH identifier
0929.62071

Subjects
Primary: 62F25
Secondary: 62H99: None of the above, but in this section

Citation

Mathew, Thomas; Sharma, Manoj Kumar; Nordström, Kenneth. Tolerance regions and multiple-use confidence regions in multivariate calibration. Ann. Statist. 26 (1998), no. 5, 1989--2013. doi:10.1214/aos/1024691366. https://projecteuclid.org/euclid.aos/1024691366

References

• Brown, P. J. (1982). Multivariate calibration (with discussion). J. Roy. Statist. Soc. Ser. B 44 287-321.
• Brown, P. J. (1993). Measurement, Regression, and Calibration. Oxford Univ. Press.
• Brown, P. J. and Sundberg, R. (1987). Confidence and conflict in multivariate calibration. J. Roy. Statist. Soc. Ser. B 49 46-57.
• Carroll, R. J., Spiegelman, C. H. and Sacks, J. (1988). A quick and easy multiple-use calibration-curve procedure. Technometrics 30 137-141.
• Davis, A. W. and Hay akawa, T. (1987). Some distribution theory relating to confidence regions in multivariate calibration. Ann. Inst. Statist. Math. 39 141-152.
• Eaton, M. L. (1983). Multivariate Statistics, A Vector Space Approach. Wiley, New York.
• Eberhardt, K. R. and Mee, R. W. (1994). Constant-width calibration intervals for linear regression. J. Quality Technology 26 21-29.
• Fujikoshi, Y. and Nishii, R. (1984). On the distribution of a statistic in multivariate inverse regression analysis. Hiroshima Math. J. 14 215-225.
• Lee, Y.-T. and Mathew, T. (1998). Tolerance intervals and calibration in linear regression. Unpublished manuscript.
• Mathew, T. and Kasala, S. (1994). An exact confidence region in multivariate calibration. Ann. Statist. 22 94-105.
• Mathew, T. and Nordstr ¨om, K. (1997). Inequalities for the probability content of a rotated ellipse and related stochastic domination results. Ann. Appl. Probab. 7 1106-1117.
• Mathew, T. and Zha, W. (1996). Conservative confidence regions in multivariate calibration. Ann. Statist. 24 707-725.
• Mathew, T. and Zha, W. (1997). Multiple use confidence regions in multivariate calibration. J. Amer. Statist. Assoc. 92 1141-1150.
• Mee, R. W. and Eberhardt, K. R. (1996). A comparison of uncertainty criteria for calibration. Technometrics 38 221-229.
• Mee, R. W., Eberhardt, K. R. and Reeve, C. P. (1991). Calibration and simultaneous tolerance intervals for regression. Technometrics 33 211-219.
• Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
• Oman, S. D. (1988). Confidence regions in multivariate calibration. Ann. Statist. 16 174-187.
• Oman, S. D. and Wax, Y. (1984). Estimating fetal age by ultrasound measurements: an example of multivariate calibration. Biometrics 40 947-960.
• Osborne, C. (1991). Statistical calibration: a review. Internat. Statist. Rev. 59 309-336.
• Scheff´e, H. (1973). A statistical theory of calibration. Ann. Statist. 1 1-37.
• Sharma, M. K. (1996). Multiple use and simultaneous confidence regions in calibration. Ph.D. dissertation, Univ. Mary land, Baltimore County.
• Sundberg, R. (1994). Most modern calibration is multivariate. In Seventeenth International Biometric Conference, Hamilton, Canada 395-405.