Abstract: 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”.
Keywords and phrases: calibration, confidence interval, Fieller’s theorem, experimental design.
