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.
Being able to appropriately model the distribution of rare events is central in many areas, such as in environmental sciences, to better manage their risks and societal impacts. The two key probabilistic models suited for the analysis of extreme events are the Generalised Extreme-Value (GEV) distribution, to model the maxima over disjoint blocks of observations, and the Generalised Pareto (GP) distribution, to model observations above a high threshold (Coles, 2001). The estimation of their parameters is however a recurrent challenge due to their high sensibility to outliers, which is linked to the restrictive sample size inherent at the nature of the problem, and because of the strong interactions between the different parameters. We propose in this article a reparametrisation of the GEV distribution based on the framework developed in Cox and Reid (1987), whose main advantage is the reduction of correlation between parameter estimates. Similar reparametrisations have been successfully applied to the GP distribution (Chavez-Demoulin and Davison, 2005) and the Weibull distribution (Hartmann and Vanhatalo, 2019). In both cases, the orthogonal parametrisation has demonstrated improved performance, for example, in terms of faster convergence of MCMC algorithms for approximation of posterior distributions in a Bayesian setting (see, respectively, Moins, 2023;Tanskanen, 2018). Deriving these orthogonal parametrisations requires knowing the elements of the Fisher information matrix for the distribution: for extreme values distributions these results are scattered and not all fully available in the literature. In this article we collate all these results in a unique place, providing a useful reference for those seeking to perform inference for extremes.
The article is organized as follows. Section 2 presents background notions of Extreme-Value Theory and the orthogonal parametrization exposed in (Cox and Reid, 1987). Section 3 proposes the different reparametrizations for the Gumbel and the 2-parameter GEV distribution. Section 4 illustrates the main properties of the reparametrisations through a simulation study.
The probability density function of the GEV distribution is given by
where pµ, σ, ξq P Rˆs0, 8rˆR denote the location, scale and shape parameters, respectively, and for x P rµ ´σ{ξ, 8r if ξ ą 0 and for x Ps ´8, µ ´σ{ξs if ξ ă 0. The limit case when ξ Ñ 0 corresponds to the Gumbel distribution whose density is given by
for x P R.
In this paper, the terminology orthogonal parametrisation refers to the reparametrisation presented in Cox and Reid (1987) who propose a general framework yielding orthogonality properties in the Fisher information matrix. For more information about this procedure, and its convenient properties, the reader is invited to consult the original paper, and its rich discussion. Here we recall only some basic notions.
Suppose that initially a log-likelihood l ˚is specified in terms of parameters pψ, θ 1 , …, θ p q :" pψ, θq P R ˆRp . The first parameter ψ is typically the parameter of interest which should remain unchanged under reparametrisation. The other parameters are sometimes referred to as nuisance parameters, highlighting the fact that the main object of the inference is ψ. A reparametrisation pψ, λq Þ Ñ θpψ, λq is defined through the relation l ˚pψ, θpψ, λqq " lpψ, λq, where l denotes the log-likelihood expressed in the new parametrisation. This parametrisation is called orthogonal if, for all j P 1, …, p, the Fisher cross-information between ψ and λ j is zero:
If the transformation θ has a nonzero Jacobian, then, by the chain rule, this implies that, for all j P t1, …, pu
and where the i ˚’s denote the elements of the Fisher information matrix in the original settings.
Remark 2.1. The notion of orthogonality considered here, in the sense of Cox and Reid (1987), is only local. That is, after the reparametrisation, the parameter ψ becomes orthogonal to the new parameters λ 1 , . . . , λ p , however there is no guarantee that the parameters λ 1 , . . . , λ p are mutually orthogonal.
We present our main contributions in this section. Our objective is to propose convenient orthogonal reparametrisations for the GEV distribution. Since the derivation of a general orthogonal reparametrisation for the full three-parameter distribution appears to be intractable (see Remark 3.2), we focus instead on two special two-parameter cases. In Section 3.1, we derive reparametrisations for the Gumbel distribution, which is the limit distribution of the GEV distribution when ξ Ñ 0. In Section 3.2, we propose to study the special case when ξ ‰ 0 and the lower or upper finite bound of the distribution is known. This assumption can be justified by physical considerations, as discussed, in, e.g., Zhang and Boos (2023), in the context of temperature extremes. Knowledge of this bound reduces the model to a two-parameter GEV distribution (1), for which an explicit orthogonal reparametrisa
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