Note on the correction of negative measured values if the measurand is positive or 0 with known probability

Abstract

Often the amount of a substance or the activity of radionuclides in a sample is measured indirectly as the difference between signal and noise, i.e. the difference between the measured value obtained at the sample and that obtained at a sample not containing the substance or the radionuclides (blank sample). The difference can be negative, especially if the concentration or the activity is low. Since a negative measurement result for a nonnegative measurand does not make sense, measured values must be corrected to nonnegative measurement results. We deal with the situation in which it is known that the measurand is 0 with a probability $p_0$ that is a priori known, and that the standard deviation $\sigma$ of the measurement is known. For this case Korun, Vodenik and Zorko extend an earlier paper by Korun and Zorko and derive the mean of the posterior distribution as a Bayesian estimator of the measurand. We offer an estimator that is based on the posterior probability $\hat{p}_0$ of the measurand being 0. If $\hat{p}_0 > 1 - \hat{p}_0$ it is 0 and otherwise the mode of the posterior distribution. This estimate is easier to calculate and less biased than that of Korun et al.