On the Genuine Replicate Design for Statistical Calibration

Let us consider the problem involving measurements on one variable taken using two instruments or methods: one is very precise but slow and expensive and another is very quick, cheap but less precise. We are interested in estimating the value of precise measure X corresponding to the value of the imprecise measure Y. This is a typical univariate calibration problem. Usually, a calibration experiment consists in running a series of experiments to obtain data on Y for fixed values of X. In the absolute calibration we assume that the measure X is without error and the measure Y is affected by the experimental error. Generally, the practical “standard design” for the calibration experiment involves the use of n distinct values of X and the measure of the corresponding experimental values of Y. We propose to consider, in the first stage of calibration procedure, a “genuine replicate design” to estimate the calibration curve. Specifically, let x be a vector of n fixed known values of X, enough representative of the X ‘s range. For each x we run a completely randomized experiment to obtain measurements on Y. Moreover, we have to replicate m times this type of experiment, with Data are taken in different and completely randomized experiments. Note that they are not just repeated readings of Y but they are genuine replicates and they provide an estimate of the pure error. In this paper, we investigate the statistical properties of the point and interval classical calibration estimator under a “genuine replicate design”. More specifically, assuming a linear calibration model, we are interested to show, using algebraic and geometric considerations, that the corresponding exact and non symmetric confidence intervals obtained using this experimental design are, under a general condition, shorter, i.e., more precise, than those obtained under the “standard design”.