Multiple Use Confidence Intervals for a Univariate Statistical Calibration

The statistical calibration problem treated here consists of constructing the interval estimates for future unobserved values of a univariate explanatory variable corresponding to an unlimited number of future observations of a univariate response variable. An interval estimate is to be computed for a value x of an explanatory variable after observing a response Y_x by using the same calibration data from a single calibration experiment, and it is called the multiple use confidence interval. It is assumed that the normally distributed response variable Y_x is related to the explanatory variable x through a linear regression model, a polynomial regression is probably the most frequently used model in industrial applications. Construction of multiple use confidence intervals (MUCI’s) by inverting the tolerance band for a linear regression has been considered by many authors, but the resultant MUCI’s are conservative. A new method for determining MUCI’s is suggested straightforward from their marginal property assuming a distribution of the explanatory variable. Using simulations, we show that the suggested MUCI’s satisfy the coverage probability requirements of MUCI’s quite well and they are narrower than previously published. The practical implementation of the proposed MUCI’s is illustrated in detail on an example.