Multivariate extreme-value models

The limit distributions of coordinate-wise maxima do not form a parametric family. There are parametric models, which can be fitted by maximum likelihood. The fit should be checked by statistical tests (this question is not treated here).

There are different approaches:

The MEVD assuming unit Fréchet margins can be written as \[ G_{\text{Fréchet}} (t_1,\dots,t_d)= \exp \Bigl(-V (t_1,\dots,t_d) \Bigr), \] with \[ V (t_1,\dots,t_d)=\int_{\mathcal{S}_d} \bigvee_{i=1}^d\Big(\frac{w_i}{t_i}\Big) S(d\mathbf{w}), \label{expmeas} \] where \(S\) is a finite measure on the \(d\)-dimensional simplex \(\mathcal{S}_d\), satisfying the equations \[ \int_{\mathcal{S}_d} w_i S(d\mathbf{w})=1\; \text{ for } i=1,...,d, \] In the bivariate setting alternative formulae can be given. Let \(G\) be a bivariate d.f. with marginals \(G_i\). Then \[-\log G(\mathbf{x}) = \ell\{-\log G_1(x_1) ,-\log G_2(x_2) \}, \qquad \mathbf{x} \in \mathbb{R}^2\] The dependence function \(A(t)=l(1-t,t), \qquad t \in [0,1]\) satisfies the following three properties:

Parametric Models

The most popular models:

BGPD II is not absolutely continuous in the asymmetric case.

New asymmetric models

Let \(A\) be an arbitrary dependence function, \(\Psi(t):[0,1] \rightarrow [0,1]\) be strictly monotonic with \(\Psi(0)=0\) and \(\Psi(1)=1\).

\(A_{\Psi}(t)=A(\Psi(t))\) defines a dependence function iff

Let \(\Psi(t)=t+f(t)\), e.g. \(f_{\psi_1,\psi_2}(t)=\psi_1[t(1-t)]^{\psi_2}\), where \(\psi_1 \in \mathbb{R}\) and \(\psi_2 \geq 2\) are the asymmetry parameters,


Other forms of \(f\) are also possible, like a two-parameter family of polynomials over\([0,1]\) with a root in \(0<p<1\).

Bivariate threshold models

Bivariate (multivariate) threshold models can be defined in two different ways.

If we claim exceedance in all of the coordinates (BGPD I), we usually get simpler models, with nice properties (marginals are univariate GPD etc.), but we may use less data.

If we use all data that exceed the threshold \({\mathbf u}\) in at least one coordinate, we get the {BGPD II} model, [1].

This approach can be formulated as follows. Let \(\mathbf{Y}=(Y_1,...,Y_d)\) denote a random vector, \(\mathbf{u}=(u_1,...,u_d)\) be a suitably high threshold vector and \(\mathbf{X} = \mathbf{Y}-\mathbf{u} = (Y_1-u_1,...,Y_d-u_d)\) be the vector of exceedances. Then the multivariate generalized Pareto distribution (MGPD) for the \(\mathbf{X}\) exceedances can be written by a MEVD \(G\) with non-degenerate margins as \[ H({\mathbf x}) =\frac{1}{\log G\bigl(0,\dots,0 \bigr)} \log \frac{G\bigl(x_1,\dots,x_d \bigr)}{G\bigl(x_1 \wedge 0,\dots,x_d \wedge 0 \bigr)} , \label{mgpd} \] where \(0<G(0,\dots,0)<1\). This definition provides a model for observations that are extreme in at least one component. We may switch to unit Fréchet margins by the transformation \begin{eqnarray} t_i(x_i) = \frac{-1}{\log \mathcal{G}_{\xi_i,\mu_i,\sigma_i} ( x_i )}= (1 + \xi_i (x_i - \mu_i ) / \sigma_i)^{1 / \xi_i}, \label{transf} \end{eqnarray}

with \(1 + \xi_i ( x_i - \mu_i ) / \sigma_i > 0\) and \(\sigma_i > 0\) for \(i=1,...,d\).


We have used 100 years of daily flood peaks data from the Danube:

The \(\Psi\)-logistic model follows the odd shape of the data extremely well.


Zempléni, A. and Rakonczai, P. (2011). New bivariate threshold models with hydrological applications. Environmental Risk and Extreme Events, Ascona.

Rakonczai, P. and Zempléni, A. (2012). Bivariate generalized Pareto distribution in practice: models and estimation. Environmetrics, 23, 3, p. 219-227.