Abstract
The multivariate calibration problem deals with inference concerning an unknown value of a covariate vector based on an observation on a response vector. Two distinct scenarios are considered in the multivariate calibration problem: controlled calibration where the covariates are non-stochastic, and random calibration where the covariates are random. Under controlled calibration, a problem of interest is the computation of a confidence region for the unknown covariate vector. Under random calibration, the problem is that of computing a prediction region for the covariate vector. Assuming the standard multivariate normal linear regression model, rectangular confidence and prediction regions are derived using a parametric bootstrap approach. Numerical results show that the regions accurately maintain the coverage probabilities. The results are illustrated using examples. The regions currently available in the literature are all ellipsoidal, and this work is the first attempt to derive rectangular regions.
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Atkins MS, Shernoff ES, Frazier SL, Schoenwald SK, Cappella E, Marinez-Lora A, Mehta TG, Lakind D, Cua G, Bhaumik R, Bhaumik D (2015) Redesigning community mental health services for urban children: supporting schooling to promote mental health. J Consult Clin Psychol 83:839–852
Bellio R (2003) Likelihood methods for controlled calibration. Scand J Stat 30:339–353
Benton D, Krishnamoorthy K, Mathew T (2003) Inferences in multivariate-univariate calibration problems. Statistician 52:15–39
Bhaumik DK, Nordgren R (2019) Prediction and calibration for multiple correlated variables. J Multivariat Anal 173:313–327
Brown PJ (1982) Multivariate calibration. J R Stat Soc B 44:287–321
Brown PJ (1993) Measurement, Regression, and Calibration. Oxford University Press
Brown PJ, Sundberg R (1987) Confidence and conflict in multivariate calibration. J R Stat Soc B 49:46–57
Davis A, Hayakawa T (1987) Some distribution theory relating to confidence regions in multivariate calibration. Ann Inst Stat Math 39:141–152
Devijver E, Perthame E (2018) Prediction regions through inverse regression. Retrieved from: arxiv:1807.03184v1. Retrieved 31 Mar 2020
Fujikoshi Y, Nishii R (1984) On the distribution of a statistic in multivariate inverse regression analysis. Hiroshima Math J 14:215–225
Gleser LJ, Hwang JT (1987) The nonexistence of 100\(\left(1-\alpha \right)\)% confidence sets of finite expected diameter in errors-in-variables and related models. Ann Stat 15:1351–1362
Krishnamoorthy K, Mathew T (2009) Statistical Tolerance Regions: Theory, Applications and Computation. John Wiley & Sons, Applications and Computation
Lütkepohl H, Staszewska-Bystrova A, Winker P (2020) Constructing joint confidence bands for impulse response functions of VAR models - a review. Econom Stat 13:69–83
Mathew T, Kasala S (1994) An exact confidence region in multivariate calibration. Ann Stat 22:94–105
Mathew T, Zha W (1996a) Conservative confidence regions in multivariate calibration. Ann Stat 24:707–725
Mathew T, Zha W (1996b) Multiple use confidence regions in multivariate calibration. J Am Stat Assoc 92:1141–1150
Mathew T, Sharma MK, Nordström K (1998) Tolerance regions and multiple use confidence regions in multivariate calibration. Ann Stat 26(5):1989–2013
Montiel Olea J, Luis J, Plagborg-Moller M (2019) Simultaneous confidence bands: theory, implementation, and an application to SVARs. J Appl Econom 34:1–17
Oman SD (1988) Confidence regions in multivariate calibration. Ann Stat 16:174–187
Osborne C (1991) Statistical calibration: a review. Int Stat Rev 59:309–336
Sundberg R (1994) Most modern calibration is multivariate. 17th International Biometric Conference. Hamilton, Canada, pp 395–405
Sundberg R (1999) Multivariate calibration - direct and indirect regression methodology (with discussion). Scand J Stat 26:161–207
Wolf M, Wunderli D (2015) Bootstrap joint prediction regions. J Time Series Anal 36:352–376
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Lucagbo, M.D., Mathew, T. Rectangular Confidence Regions and Prediction Regions in Multivariate Calibration. J Indian Soc Probab Stat 23, 155–171 (2022). https://doi.org/10.1007/s41096-022-00116-7
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DOI: https://doi.org/10.1007/s41096-022-00116-7