7.6.1.2. The normality hypothesis

If we collect all the information about the system described so far, the problem can be summarised as

\[R^{\rm exp}_j - R^{\rm code}_j = (R^{\rm exp}_j - R^{\rm real}_j) + (R^{\rm real}_j - R^{\rm code}_j) = e_j + \sum_{i=1}^{q}\dfrac{\partial R^{\rm code}_j}{\partial \alpha_i} \times \alpha_{j,i}\]

In this expression, one can discuss the different contributions:

• \(R^{\rm exp}_j - R^{\rm code}_j\) and \(\dfrac{\partial R^{\rm code}_j}{\partial \alpha_i}\) are known.

• \(e_j\) is a realisation of \(\mathcal{N}(0,(\sigma^{\rm exp})^{2})\) where \(\sigma^{\rm exp}\) is also known.

• The \(\alpha_{j,i}\) are unknown. The only available information comes from their statistical properties: their bias \(b_i\) and their standard deviation \(\sigma_i\).

Several solutions are possible for the vector \(\alpha\), leading to a needed choice among them. The criterion chosen to do so is the maximum likelihood estimation, which requires a hypothesis on the form of the probability distribution followed by the \(\alpha_i\) parameters. The normality assumption is then adopted.