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

5.5 Understanding Rare Probabilities and Extreme Value Statistical Modelling

This section discusses the particular issues raised by the handling of rare risk measures from a statistical point of view. It will first come back to the concept of extreme value theory introduced in Chapter 4, Section 4.3.5 to underline both its advantages and limitations in the handling of limited datasets. Further, there is discussion of the sensitivity of extremely low probabilities to the choices made in the uncertainty model and the associated questions on the level of information that is truly shown.

5.5.1 The Issue of Extrapolating Beyond Data – Advantages and Limitations of the Extreme Value Theory

Chapter 4 showed that in many risk studies, the risk measure involves rare probabilities or quantiles. Start with the case where the variable or event of interest may be directly observed: rarity should then be assessed from a statistical point of view, meaning that the desired quantile reaches the limits of the available sources of information or even beyond. The section on non-parametric estimation made it clear that it becomes tricky to estimate a quantile around α = 1/n, and impossible beyond that on the sole basis of an n-sample, as shown in Figure 5.15 on flood flows.

Figure 5.15 Extrapolating beyond the empiric histogram – kernel vs. parametric model (the flood flows example: 10–1000 yearly flood return periods inferred on the basis of 30 observations in m3/s).

Beyond such an order, a parametric ...