Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods

5.2 Introducing Estimation Techniques on Independent Samples

In this section, some basic examples will be given in order to help understand the subsequent issues. Suppose that an observation sample Ξ_n = {x_j}_{j=1. . .n} is made available to estimate the uncertainty model {f_X (x|θ_X)·π(θ_X |IK, Ξ_n)}. One will start with the classical estimation procedure for the estimation of a fixed albeit unknown parameter value θ_X°, that is we are not modelling the epistemic uncertainty at this stage.

5.2.1 Estimation Basics

Statistical estimation of an uncertainty model returns to a key modelling alternative:

• infer a model belonging to a standard class of distribution shapes, such as Gaussian, Lognormal, Gumbel, and so on: this is referred to traditionally as a parametric model;
• infer a distribution adhering to data while not belonging to a previously-fixed standard shape: this is referred to traditionally as a non-parametric model.

The second option may be more accurately referred to as a ‘non-standard parametric’ approach, since it still relies on a parameterised distribution fitted to data. However, as will be seen hereafter, the distribution shape resulting from such an estimation approach enjoys much wider degrees of freedom.

As for any modelling choice, both come with a cost that will be illustrated below. Apart from a limited number of situations, it is reasonable to doubt that the distribution of uncertainty surrounding a phenomenological input, possibly the result of complex combinations ...