Inferences in multivariate-univariate calibration problems

Abstract

Summary. A normally distributed vector response variable is considered, related to a scalar explanatory variable through a linear regression model. The calibration data, i.e. data obtained on the response variable corresponding to known values of the explanatory variable, are to be used for making inferences concerning unknown values of the explanatory variable. The purpose of the paper is a detailed investigation of interval estimation and hypothesis testing problems that arise in this context. Such problems are addressed in the scenario of both single use and multiple use of the calibration data. In the single-use situation, the calibration data are used to make inferences concerning a single unknown value of the explanatory variable. In the multiple-use scenario, the calibration data are used to make inferences concerning a sequence of unknown values of the explanatory variable. A one-sided hypothesis testing problem is addressed and test procedures are developed in the context of both single use and multiple use of the calibration data. Since the test statistic has a distribution that depends on some unknown parameters, a parametric bootstrap procedure is advocated to carry out the test. The parametric bootstrap procedure can be adopted for interval estimation as well. The performance of the parametric bootstrap procedure is numerically investigated and is found to be quite satisfactory. Some examples are used to motivate the problems and to illustrate the applicability of our results. The overall conclusion is that the parametric bootstrap is a simple and satisfactory approach for making inferences in the calibration problem.