Concentration models uncertainty

When using empirical distributions, concentration model uncertainty is covered by the the inputs. I.e., concentration models can be recomputed from resampled/bootstrapped concentration data. This happens for both the univariate concentration models, being recomputed from the bootstrapped residue collections for each food and substance, and also for the samples of the sample-based approach that are re-generated from the bootstrapped samples (including the necessary steps of missing value imputation and imputation of censored values).

When parametric uncertainty is preferred over empirical bootstrapping, the parameters of the univariate concentration models fitted as a parametric distributions can be resampled parametrically.

Let \(x\) denote a random variable from the specified distribution. The log transformed variable \(y = ln(x)\) is normally distributed with mean \(\mu_{y}\) and variance \(\sigma_{y}\). The maximum likelihood estimates are \(\hat{\mu}_{y}\) and \(\hat{\sigma}_{y}\). In each bootstrap sample, values are drawn from a normal distribution where the maximum likelihood estimates are replaced by ( \(\hat{\mu}_{y}^*\), \(\hat{\sigma}_{y}^*\)).