Rectangular Confidence Regions and Prediction Regions in Multivariate Calibration

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.