Multiple Use One-Sided Hypotheses Testing in Univariate Linear Calibration

Abstract

Consider a normally distributed response variable, related to an explanatory variable through the simple linear regression model. Data obtained on the response variable, corresponding to known values of the explanatory variable (i.e., calibration data), are to be used for testing hypotheses concerning unknown values of the explanatory variable. We consider the problem of testing an unlimited sequence of one sided hypotheses concerning the explanatory variable, using the corresponding sequence of values of the response variable and the same set of calibration data. This is the situation of multiple use of the calibration data. The tests derived in this context are characterized by two types of uncertainties: one uncertainty associated with the sequence of values of the response variable, and a second uncertainty associated with the calibration data. We derive tests based on a condition that incorporates both of these uncertainties. The solution has practical applications in the decision limit problem. We illustrate our results using an example dealing with the estimation of blood alcohol concentration based on breath estimates of the alcohol concentration. In the example, the problem is to test if the unknown blood alcohol concentration of an individual exceeds a threshold that is safe for driving.