  This document presents the statistical methods used to process low-level
measurements in the presence of noise. These methods can be classical or
Bayesian. The question is placed in the general framework of the problem of
nuisance parameters, one of the canonical problems of statistical inference. By
using a simple criterion proposed by Bolstad (2007), it is possible to define
statistically significant results during a measurement process (act of
measuring in the vocabulary of metrology). This result is similar for a classic
paradigm (called ``frequentist'') or Bayesian: the presence of zero in the
interval considered (confidence or credibility). It is shown that in the case
of homoskedastic Gaussians, the commonly used results are found. The case of
Poisson distributions is then considered. In the case of heteroscedastic
Gaussians, which is that of radioactivity measurement, we can consider them as
Poisson laws in the limit of large counts. The results are different from those
commonly used, and in particular those from standards (ISO 11929). Their
statistical performances, characterized by simulation, are better and are well
verified experimentally. This is confirmed theoretically by the use of the
Neyman-Pearson lemma which makes it possible to formally determine the
statistical tests with the best performances. These results also make it
possible to understand the paradox of the possible divergence of the detection
limit. It is also formally shown that the confidence intervals thus calculated
by getting rid of the nuisance parameter according to established methods
result in the commonly used confidence interval. To our knowledge, this
constitutes the first formal derivation of these confidence intervals.This
method is based on keeping the measurement results whether they are significant
or not (not censoring them). This is recommended in several standards or
documents, is compatible with the ISO 11929 standard and is in line with recent
proposals in the field of statistics. On the other hand, all the information
necessary to determine whether a measurement result is significant or not
remains available. The conservation and restitution of all results is currently
applied in the USA. The textbook case of the WIPP incident makes it possible to
ensure favorable public perception.The implications and applications of this
method in different fields are finally discussed.
