Orthogonal parametrisations of Extreme-Value distributions

Abstract

Extreme value distributions are routinely employed to assess risks connected to extreme events in a large number of applications. They typically are two- or three- parameter distributions: the inference can be unstable, which is particularly problematic given the fact that often times these distributions are fitted to small samples. Furthermore, the distribution's parameters are generally not directly interpretable and not the key aim of the estimation. We present several orthogonal reparametrisations of the main extreme-value distributions, key in the modelling of rare events. In particular, we apply the theory developed in Cox and Reid (1987) to the Generalised Extreme-Value, Generalised Pareto, and Gumbel distributions. We illustrate the principal advantage of these reparametrisations in a simulation study.

Figures (1)

• Figure 1: Violin plots of the cross-correlation between parameter estimates for the two-parameter GEV distribution (left), the two-parameter GP distribution (middle), and the Gumbel distribution (right), under the different parametrisations considered. The cross-correlations are computed over $d=1000$ independent replications of samples of size $n=100$.

Theorems & Definitions (10)

• Remark 2.1
• Theorem 3.1: Orthogonal reparametrisation of the Gumbel distribution
• Theorem 3.2: Orthogonal reparametrisation of the two-parameter GEV distribution
• Remark 3.1: The GP distribution
• Remark 3.2: The three-parameter GEV distribution case
• proof
• proof : Proof
• proof : Proof of Theorem \ref{['thm:Gumbel']}
• proof : Proof of Theorem \ref{['thm:2-GEV']}
• proof : Proof of Remark \ref{['rem:GPD']}