Reporting measurement results of activities near the natural limit: note and extension of the article “Interpretation of measurement results near the detection limit in gamma-ray spectrometry using Bayesian statistics”

Abstract

When reporting activity measurement results near the natural limit for activities, (zero), care must be taken not to report observed values from the unfeasible region and not to report confidence intervals extending beyond the limit of the feasible range. This can be achieved by reporting best estimates, where the requirements that the best-estimate values remain in the feasible range and that the confidence intervals do not extend beyond its limit are taken into account. The use of the truncated and renormalized normal distribution, defined by the primary measurement result, as the posterior in the calculation of the best estimates fulfills both requirements, but suffers in that it introduces a bias into the measurement results reported, that is, the observed value may not lie within the interval, centered at the best-estimate value and having a width equal to twice its uncertainty, and that, consequently, the use of best estimates may lead to unacceptable results. Therefore, a posterior, comprising the Dirac delta function distribution δ(a) and the truncated normal distribution, is used for the calculation of the best estimate. It is shown that here the bias introduced is smaller; the observed value lies within the interval centered on the best-estimate value and has a width equal to twice its uncertainty, and that these best estimates perform better when used in calculations of the mean.