In this article we consider a Bayesian approach to inference in which there is a calibration relationship between measured and true quantities of interest. One situation in which this approach is useful is for unknowns in which calibration intervals are obtained. The other situation is when inference about a population is desired in which tolerance intervals are produced. The Bayesian approach easily handles a general calibration relationship, say nonlinear, with nonnormal errors. The population may also be general, say lognormal, for quantities which are nonnegative. The Bayesian approach is illustrated with three examples and implemented with the freely available WinBUGS software.Key Words: Linear Calibration, Lognormal Population, Nonlinear Calibration Models, Normal Population, Tolerance Bounds.

*By M. HAMADA Los Alamos National Laboratory, Los Alamos NM 87545
A. POHL Air Force Institute of Technology, OH 45433
C. SPIEGELMAN Texas A & M University, College Station TX 77843
J. WENDELBERGER Los Alamos National Laboratory, Los Alamos NM 87545*

**Introduction**

MEASURING devices often measure a quantity of interest indirectly and hence require a calibration that relates an observed measurement to the true quantity. Use of such a calibration arises in a number of situations. First, an unknown may be measured, and inference about its true quantity is desired. Second, a sample of unknowns may be taken, and inference about the distribution of the true quantities is the goal; e.g., the mean or some quantile may be appropriate to addressing a particular scientific problem.

The calibration relationship or model is usually determined by an experiment, the so-called calibration experiment. Consequently, there is uncertainty in the relationship that is expressed in terms of the uncertainties in the calibration model parameters. These uncertainties need to be properly accounted for in the two inference problems listed in the previous paragraph. In fact, Mulrow et al. (1988) identified the serious problem of treating predicted quantities from the calibration relationship as true quantities to obtain confidence intervals about the mean of a population as well as tolerance bounds, i.e., confidence bounds on a population quantile. In this article, we consider how a Bayesian approach naturally handles the uncertainty in the calibration model to properly make these inferences.

Previously, Hoadley (1970), Hunter and Lamboy (1981), Ghosh, Carlin, and Srivastava (1995), and Eno (1999) used a Bayesian approach to obtain a confidence interval on a single unknown for the simple linear calibration model. In this article, we also show how a Bayesian approach easily handles more complicated calibration models such as a non-linear one (Racine-Poon (1988)). The recent advances in Bayesian computation such as Markov chain Monte Carlo methods (Casella and George (1992) and Chib and Greenberg (1995)) make this practical. In fact, WinBUGS is freely available software, and we demonstrate how it can be used to implement our approach.

An outline of the article is as follows. Three examples involving biphenyl melting temperatures, traffic speeds, and element contents are first introduced. Then a Bayesian approach to these statistical problems is presented. We then apply the Bayesian approach to the three examples. Finally, we conclude the article with a short discussion.

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