High-dimensional inference using the extremal skew-t process

Abstract

Max-stable processes are a popular tool for the study of environmental extremes, and the extremal skew-t process is a general model that allows for a flexible extremal dependence structure. For inference on max-stable processes with high-dimensional data, exact likelihood-based estimation is computationally intractable. Composite likelihoods, using lower dimensional components, and Stephenson-Tawn likelihoods, using occurrence times of maxima, are both attractive methods to circumvent this issue for moderate dimensions. In this article we establish the theoretical formulae for simulations of and inference for the extremal skew-t process. We also incorporate the Stephenson-Tawn concept into the composite likelihood framework, giving greater statistical and computational efficiency for higher-order composite likelihoods. We compare 2-way (pairwise), 3-way (triplewise), 4-way, 5-way and 10-way composite likelihoods for models of up to 100 dimensions. Furthermore, we propose cdf approximations for the Stephenson-Tawn likelihood function, leading to large computational gains, and enabling accurate fitting of models in large dimensions in only a few minutes. We illustrate our methodology with an application to a 90-dimensional temperature dataset from Melbourne, Australia.

Appendices

Appendix A: Technical details

1.1 A.1 The non-central extended skew-t distribution (Beranger et al. 2017)

Definition 1

Y is a d-dimensional, non-central extended skew-t distributed random vector, denoted by $Y\sim \mathcal {S}\mathcal {T}_{d}(\mu , {\Omega }, \alpha , \tau ,\kappa ,\nu )$, if for $y\in \mathbb {R}^{d}$ it has pdf

$$ \psi_{d}(y;\mu,{\Omega},\alpha,\tau,\kappa,\nu)=\frac{\psi_{d}(y;\mu,{\Omega},\nu)} {{\Psi}\left( \frac{\tau}{\sqrt{1+Q_{\bar{\Omega}}(\alpha)}};\frac{\kappa}{\sqrt{1+Q_{\bar{\Omega}}(\alpha)}},\nu\right)} {\Psi} \left\{(\alpha^{\top} z+\tau) \sqrt{\frac{\nu + d}{\nu + Q_{\bar{\Omega}^{-1}}(z)}};\kappa,\nu+d\right\}, $$

where ψ_d(y; μ,Ω, ν) is the pdf of a d-dimensional t-distribution with location $\mu \in \mathbb {R}^{d}$, d × d scale matrix Ω and $\nu \in \mathbb {R}^{+}$ degrees of freedom, Ψ(⋅; a, ν) denotes a univariate non-central t cdf with non-centrality parameter $a\in \mathbb {R}$ and ν degrees of freedom, and $Q_{\bar {\Omega }^{-1}}(z)=z^{\top } \bar {\Omega }^{-1} z$, z = (y − μ)/ω, ω = diag(Ω)^1/2, $\bar {\Omega }=\omega ^{-1} {\Omega } \omega ^{-1}$ and $Q_{\bar {\Omega }}(\alpha )=\alpha ^{\top } \bar {\Omega }\alpha$. The associated cdf is

$$ {\Psi}_{d}(y;\mu,{\Omega},\alpha,\tau,\kappa,\nu) = \frac{{\Psi}_{d+1} \left\{ \bar{z}; {\Omega}^{\ast},\kappa^{\ast}, \nu \right\}} {{\Psi}\left( \bar{\tau}; \bar{\kappa},\nu\right)}, $$

where $\bar {z}=(z^{\top },\bar {\tau })^{\top }$, Ψ_d+ 1 is a (d + 1)-dimensional (non-central) t cdf with covariance matrix and non-centrality parameters

$$ {\Omega}^{\ast}=\left( \begin{array}{cc} \bar{\Omega} & - \delta\\ - \delta^{\top} & 1 \end{array} \right), \quad \kappa^{\ast}=\left( \begin{array}{c} 0 \\ \bar{\kappa} \end{array} \right), $$

and ν degrees of freedom, and where

$$ \delta = \left\{ 1+Q_{\bar{\Omega}}(\alpha) \right\}^{-1/2} \bar{\Omega}\alpha,\quad \bar{\kappa} = \left\{ 1+Q_{\bar{\Omega}}(\alpha) \right\}^{-1/2} \kappa, \quad \bar{\tau} = \left\{ 1+Q_{\bar{\Omega}}(\alpha) \right\}^{-1/2} \tau. $$

1.2 A.2 Parameters of the exponent function of the extremal skew-t

First of all, we define m_j+ ≡ m₊(s_j) as follows

$$ m_{j+} = {\int}_{0}^{\infty} {y}_{j}^{\nu} \phi(y_{j};{\alpha}_{j}^{\ast},{\tau}_{j}^{\ast})\mathrm{d} y_{j} = \frac{2^{(\nu-2)/2} {\Gamma}\{(\nu+1)/2\} {\Psi}({\alpha}_{j}^{\ast} \sqrt{\nu+1};-{\tau}_{j}^{\ast},\nu+1)} {\sqrt{\pi}{\Phi}[\tau\{1+Q_{\bar{\Omega}}(\alpha)\}^{-1/2}]}, $$

where $\alpha ^{\ast }_{j}$ and $\tau _{j}^{\ast }$ are respectively the marginal shape and extension parameters and $Q_{\bar {\Omega }}(\alpha ) = \alpha ^{\top } \bar {\Omega } \alpha$. This then allows us to obtain the exponent function of the extremal skew-t given in Eq. 4 as the sum of (d − 1)-dimensional non-central extended skew-t distribution with correlation matrix $\bar {\Omega }_{i}^{\circ }=\omega _{I_{i}I_{i}\cdot i}^{-1} \bar {\Omega }_{I_{i}I_{i}\cdot i} \omega _{I_{i}I_{i}\cdot i}^{-1}$, where $\omega _{I_{i}I_{i}\cdot i}=\text {diag}(\bar {\Omega }_{I_{i}I_{i}\cdot i})^{1/2}$, $\bar {\Omega }_{I_{i}I_{i}\cdot i}=\bar {\Omega }_{I_{i}I_{i}}-\bar {\Omega }_{I_{i}i}\bar {\Omega }_{iI_{i}}$, I = {1,…, d}, I_i = I∖i, $\bar {\Omega }=\omega ^{-1} {\Omega } \omega ^{-1}$, ω = diag(Ω)^1/2. The shape parameter is $\alpha _{i}^{\circ }=\omega _{I_{i}I_{i}\cdot i} \alpha _{I_{i}} \in \mathbb R^{d-1}$, the extension parameter $\tau _{j}^{\circ }=(\bar {\Omega }_{jI_{j}}\alpha _{I_{j}}+\alpha _{j})(\nu +1)^{1/2} \in \mathbb R$, the non-centrality $\kappa ^{\circ }_{j}=-\{1+Q_{\bar {\Omega }_{I_{j}I_{j}\cdot j}}(\alpha _{I_{j}})\}^{-1/2}\tau \in \mathbb R$$Q_{\bar {\Omega }_{I_{j}I_{j}\cdot j}}(\alpha _{I_{j}}) = \alpha _{I_{j}}^{\top } \bar {\Omega }_{I_{j}I_{j}\cdot j} \alpha _{I_{j}}$, and the degrees of freedom ν + 1.

1.3 A.3 Proof of proposition 1

Lemma 1

The finite d-dimensional distribution of the random process $(W(s) / W(s_{0}))_{s \in \mathcal {S}}$ under the transformed probability measure $\widehat {\Pr } = \{ W(s_{0})\}_{+}^{\nu } / m_{+}(s_{0}) \mathrm {d}\Pr$ is equal to the distribution of a non-central extended skew-t process with mean μ_d, d × d scale matrix $\hat {\Sigma }_{d}$, slant vector $\hat {\alpha }_{d}$, extension τ_d, non-centrality κ_d = −τ and ν_d = ν + 1 degrees of freedom, given by

$$ \begin{array}{@{}rcl@{}} \mu_{d} &=& {\Sigma}_{d;0}, \quad \hat{\Sigma}_{d} = \frac{{\Sigma}_{d} - {\Sigma}_{d;0} {\Sigma}_{0;d} }{\nu +1 }, \quad \hat{\alpha}_{d} = \sqrt{\nu +1} \hat{\omega} \omega_{d}^{-1} \alpha,\\ \tau_{d} &=& (\alpha_{0} + {\Sigma}_{0;d} \omega_{d}^{-1} \alpha ) \sqrt{\nu + 1}, \end{array} $$

and where Σ_d = (K(x_i, x_j))_{1≤i, j, ≤d}, ${\Sigma }_{d;0} = {\Sigma }_{0;d}^{\top } = (K(x_{0}, x_{i})) {1 \leq i \leq d}$, α = (α₁, … , α_d), ω_d = diag(Σ_d)^1/2 and $\hat {\omega } = \text {diag}(\hat {\Sigma }_{d})^{1/2}$.

Proof of Lemma 1

The proof runs along the same lines as the proof of Lemma 2 in the supplementary material of Dombry et al. (2016). We consider finite dimensional distributions only. Let d ≥ 1 and $s_{1}, \ldots , s_{d} \in \mathcal {S}$. We assume that the covariance matrix $\tilde {\Sigma } = (K(s_{i}, s_{j}))_{0 \leq i,j \leq d}$ is non singular so that (W(s_i))_0≤i≤d has density

$$ \begin{array}{@{}rcl@{}} \tilde{g}(y) &=& (2 \pi)^{-(d+1)/2} \det (\tilde{\Sigma})^{-1/2} \exp\left\{ -\frac{1}{2} y^{\top} \tilde{\Sigma}^{-1} y \right\} \frac{\Phi(\tilde{\alpha}^{\top} \tilde{\omega}^{-1} y + \tau)}{\Phi(\tau/ \sqrt{1 + Q_{\tilde{\bar{\Sigma}}^{-1}}(\tilde{\alpha})}) },\\ &&y \in \mathbb {R}^{d+1}, \end{array} $$

where $\tilde {\alpha } = (\alpha _{0}, \alpha _{1}, \ldots , \alpha _{d} )$, $Q_{\tilde {\bar {\Sigma }}^{-1}}(\tilde {\alpha }) = \tilde {\alpha }^{\top } \tilde {\bar {\Sigma }} \tilde {\alpha }$, $\tilde {\bar {\Sigma }} = \tilde {\omega }^{-1} \tilde {\Sigma } \tilde {\omega }^{-1}$ and $\tilde {\omega } = \text {diag}(\tilde {\Sigma })^{1/2}$. Setting z = (y_i/y₀)_1≤i≤d, for all Borel sets $A_{1}, \ldots , A_{d} \subset \mathbb {R}$,

$$ \begin{array}{@{}rcl@{}} \widehat{\Pr} \left\{ \frac{W(s_{i})}{W(s_{0})} \in A_{i}, i=1, \ldots, d \right\}&=& {\int}_{\mathbb {R}^{d+1}} \mathbb{I} \{ y_{i}/y_{0} \in A_{i}, i=1, \ldots, d \} \frac{(y_{0})_{+}^{\nu} \tilde{g}(y)}{m_{0+}} \mathrm{d} y \\ &=&{\int}_{\mathbb {R}^{d}} \mathbb{I} \{ z_{i} \in A_{i}, i=1, \ldots, d \} \left\{ {\int}_{0}^{\infty} \frac{(y_{0})_{+}^{\nu} \tilde{g}((y_{0}, y_{0} z)^{\top})}{m_{0+}} {y_{0}^{d}} \mathrm{d} y_{0} \right\} \mathrm{d} z. \end{array} $$

We deduce that under $\widehat {\Pr }$ the random vector (W(s_i)/W(s₀))_1≤i≤d has density

$$ \begin{array}{@{}rcl@{}} g(z)&=& {\int}_{0}^{\infty} \frac{y_{0}^{d+\nu}}{m_{0+}} \tilde{g}((y_{0}, y_{0} z)^{\top}) \mathrm{d} y_{0} \\ &=& \frac{(2\pi)^{-(d+1)/2} \det(\tilde{\Sigma})^{-1/2}} {m_{0+} {\Phi}(\tau/ \sqrt{1 + Q_{\tilde{\bar{\Sigma}}^{-1}}(\tilde{\alpha})})} {\int}_{0}^{\infty} y_{0}^{d+\nu} \exp\left\{ -\frac{1}{2} \tilde{z}^{\top} \tilde{\Sigma}^{-1} \tilde{z} {y_{0}^{2}} \right\} {\Phi}(\tilde{\alpha}^{\top} \tilde{\omega}^{-1} (y_{0}, y_{0} z)^{\top} + \tau) \mathrm{d} y_{0}, \end{array} $$

where $\tilde {z} = (1, z)^{\top }$. Through the change of variable $u = \left (\tilde {z}^{\top } \tilde {\Sigma }^{-1} \tilde {z} \right )^{1/2} y_{0}$, we obtain

$$ \begin{array}{@{}rcl@{}} {\int}_{0}^{\infty} y_{0}^{d+\nu} &&\exp \left\{ -\frac{1}{2} \tilde{z}^{\top} \tilde{\Sigma}^{-1} \tilde{z} {y_{0}^{2}} \right\} {\Phi}(\tilde{\alpha}^{\top} \tilde{\omega}^{-1} (y_{0}, y_{0} z) + \tau) \mathrm{d} y_{0} \\ &=& \sqrt{2\pi} \left( \tilde{z} \tilde{\Sigma}^{-1} \tilde{z} \right)^{-\frac{d + \nu +1}{2}} {\int}_{0}^{\infty} u^{d + \nu} \phi(u) {\Phi} \left( \frac{\tilde{\alpha}^{\top} \tilde{\omega}^{-1} \tilde{z} } { \sqrt{ \tilde{z} \tilde{\Sigma}^{-1} \tilde{z} } } u + \tau \right) \mathrm{d} u \\ &=& \left( \tilde{z} \tilde{\Sigma}^{-1} \tilde{z} \right)^{-\frac{d_{\nu}}{2}} 2^{\frac{\nu - 3}{2}} \pi^{-\frac{d+2}{2}} {\Gamma}\left( \frac{d_{\nu}}{2}\right) {\Psi} \left( \frac{ \tilde{\alpha}^{\top} \tilde{\omega}^{-1} \tilde{z} } {\sqrt{ \tilde{z} \tilde{\Sigma}^{-1} \tilde{z} } } \sqrt{d_{\nu}}; -\tau, d_{\nu} \right), \end{array} $$

where α = (α₁,…, α_d), d_ν = d + ν + 1 and Ψ(⋅; κ, ν) is the cdf of the non-central t distribution with non-centrality parameter κ and ν degrees of freedom. Thus applying the definition of m₀₊, ${\alpha }^{\ast }_{0}$ and ${\tau }^{\ast }_{0}$ from Beranger et al. (2017) we get

$$ g(z) = \frac{\pi^{-d/2} \det(\tilde{\Sigma})^{-1/2} } { {\Psi}(\alpha^{\ast}_{0} \sqrt{\nu +1}; -\tau^{\ast}_{0}, \nu +1) } \left( \tilde{z} \tilde{\Sigma}^{-1} \tilde{z} \right)^{-\frac{d_{\nu}}{2}} \frac{{\Gamma}\left( \frac{d_{\nu}}{2}\right)}{{\Gamma}\left( \frac{\nu + 1}{2}\right)} {\Psi} \left( \frac{\tilde{\alpha}^{\top} \tilde{\omega}^{-1} \tilde{z}} {\sqrt{\tilde{z} \tilde{\Sigma}^{-1} \tilde{z}} } \sqrt{d_{\nu}}; -\tau, d_{\nu} \right), $$

and the block decomposition $\tilde {\Sigma } = \left (\begin {array}{cc} 1 & {\Sigma }_{0;d} \\ {\Sigma }_{d;0} & {\Sigma }_{d} \end {array} \right )$, allows us to write

$$ g(z) = \frac{\psi_{d}(z; \mu_{d}, \hat{\Sigma}_{d}, \nu + 1 ) {\Psi} \left( \frac{\alpha_{0} + \alpha^{\top} \omega_{d}^{-1} z} {\sqrt{ \tilde{z} \tilde{\Sigma}^{-1} \tilde{z} } } \sqrt{d_{\nu}} ; -\tau, d_{\nu} \right)} { {\Psi}(\alpha_{0}^{\ast} \sqrt{\nu +1}; -\tau_{0}^{\ast}, \nu +1)}, $$

where μ_d = Σ_d;0, $\hat {\Sigma }_{d} = ({\Sigma }_{d} - {\Sigma }_{d;0} {\Sigma }_{0;d}) / (\nu + 1)$ and ω_d = diag(Σ_d)^1/2. Noting that

$$ \begin{array}{@{}rcl@{}} {\Psi}(\alpha^{\ast}_{0} \sqrt{\nu +1}; -\tau^{\ast}_{0}, \nu +1) &=& {\Psi} \left( \frac{\tau_{d}} {\sqrt{1 + \hat{\alpha}_{d}^{\top} \hat{\bar{\Sigma}}_{d} \hat{\alpha}_{d} }}; \frac{-\tau}{\sqrt{1 + \hat{\alpha}_{d}^{\top} \hat{\bar{\Sigma}}_{d} \hat{\alpha}_{d} }}; \nu + 1 \right) \\ {\Psi} \left( \frac{\alpha_{0} + \alpha^{\top} \omega_{d}^{-1} z } {\sqrt{ \tilde{z} \tilde{\Sigma}^{-1} \tilde{z} } } \sqrt{d_{\nu}} ; -\tau, d_{\nu} \right) &=& {\Psi} \left( \frac{ \hat{\alpha}_{d} z^{\prime} + \tau_{d} } {\sqrt{\nu + 1 + z^{{\prime}{\top}} \hat{\bar{\Sigma}}_{d}^{-1} z^{\prime} }} \sqrt{d_{\nu} }; - \tau, d_{\nu} \right), \end{array} $$

where Ψ(⋅; κ, ν) denotes the cdf of the univariate non-central t distribution with non-centrality κ and ν degrees of freedom, $z^{\prime } = \hat {\omega }^{-1} (z - {\Sigma }_{d;0})$, $\hat {\alpha }_{d} = \sqrt {\nu + 1} \hat {\omega } \omega _{d}^{-1} \alpha$, $\tau _{d} = (\alpha _{0} + \alpha ^{\top } {\Sigma }_{d;0}) \sqrt {\nu + 1 }$, $\hat {\bar {\Sigma }}_{d} = \hat {\omega }^{-1} \hat {\Sigma } \hat {\omega }^{-1}$, $\hat {\omega } = \text {diag}(\hat {\Sigma })^{1/2}$ which leads us to the conclusion that g(z) is the density of a non-central extended skew-t distribution with parameters μ_d = Σ_d;0, $\hat {\Sigma }_{d}$, $\hat {\alpha }_{d}$, τ_d and κ_d = −τ. □

In order to prove Proposition 1, let $\mathcal {C}_{+} = \mathcal {C} \{\mathcal {S}, [0, \infty ]\}$ denote the space of continuous non-negative functions on $\mathcal {S}$, and σ represent the distribution of the $\{W(s_{i})\}_{+}^{\nu } / m_{+}$, and consider the set $A = \{ f \in \mathcal {C}_{0} : f(s_{1}) \in A_{1}, \ldots , f(s_{d}) \in A_{d} \}.$ Then by (Dombry and Eyi-Minko 2013, Proposition 4.2) we have

$$ \begin{array}{@{}rcl@{}} P_{s_{0}}(A)&=&{\int}_{\mathcal{C}} \mathbb{I} \{ f/f(s) \in A \} f(s) \sigma(\mathrm{d} f) \\ &=& \mathbb{E} \left[\{W(s_{0})\}_{+}^{\nu} / m_{0+} \mathbb{I} \left\{ \frac{m_{0+} \{W(s_{i})\}_{+}^{\nu}}{m_{i+} \{ W(s_{0})\}_{+}^{\nu}} \in A_{i}; i = 1, \ldots, d\right\} \right] \\ &=& \widehat{\Pr} \left\{ \frac{m_{0+} \{W(s_{i})\}_{+}^{\nu}}{m_{i+} \{ W(s_{0})\}_{+}^{\nu}} \in A_{i}; i = 1, \ldots, d \right\} \\ &=& \Pr \left\{ \frac{m_{0+}}{m_{i+}} (T_{i})_{+}^{\nu} \in A_{i}; i = 1, \ldots, d \right\}, \end{array} $$

where T = (T₁,…, T_d), T_i = W(s_i)/W(s₀) which, from Lemma 1, is distributed as

$$ T \sim {\Psi}_{d} \left( {\Sigma}_{d;0}, \frac{{\Sigma}_{d} - {\Sigma}_{d;0} {\Sigma}_{0;d} }{\nu +1 }, \hat{\alpha}_{d}, \tau_{d}, - \tau, \nu + 1 \right). $$

1.4 A.4 Proof of Proposition 2

The following Lemma is required in order to complete the proof.

Lemma 2

Under the assumptions of Proposition 2, the intensity function of the extremal skew-t is

$$ \lambda_{\boldsymbol{s}} (\boldsymbol{v}) = \frac{ 2^{(\nu-2)/2} \nu^{-d+1} {\Gamma} \left( \frac{d+\nu}{2} \right) {\Psi} \left( \tilde{\alpha}_{\boldsymbol{s}} \sqrt{d+\nu}; -\tau_{\boldsymbol{s}}, d+\nu \right) {\prod}_{i=1}^{d} \left( m_{i+} v_{i}^{1-\nu} \right)^{1/\nu} } { \pi^{d/2} |\bar{\Omega}_{\boldsymbol{s}}|^{1/2} Q_{\bar{\Omega}_{\boldsymbol{s}}}(\boldsymbol{v}^{\circ})^{(d+\nu)/2} {\Phi}(\tau_{\boldsymbol{s}} \{ 1 + Q_{\bar{\Omega}^{-1}_{\boldsymbol{s}}}(\alpha_{\boldsymbol{s}}) \}^{-1/2} ) }, $$

where $\boldsymbol {v}^{\circ } = (\boldsymbol {v} m_{+}(\boldsymbol {s}))^{1/\nu } \in \mathbb {R}^{d}$, $\tilde {\alpha }_{\boldsymbol {s}} = \alpha _{\boldsymbol {s}}^{\top } \boldsymbol {v}^{\circ } Q_{\bar {\Omega }_{\boldsymbol {s}}}(\boldsymbol {v}^{\circ })^{-1/2} \in \mathbb {R}$ and $\alpha _{\boldsymbol {s}} \in \mathbb {R}^{d}$.

Proof

By definition of the intensity measure (5), for all $\boldsymbol {s} \in \mathcal {S}^{d}$ and Borel set $A \subset \mathbb {R}^{d}$,

$$ {\Lambda}_{\boldsymbol{s}}(A) = {\int}_{0}^{\infty} \int\limits_{\mathbb {R}^{d}} \mathbb{I} \left\{ \zeta \boldsymbol{t}^{\nu} / m_{+}(\boldsymbol{s}) \in A \right\} g_{\boldsymbol{s}}(\boldsymbol{t}) \mathrm{d} \boldsymbol{t} \zeta^{-2} \mathrm{d} \zeta, $$

where g_s is the density of the random vector W(s), i.e. a centred extended skew normal random vector with correlation matrix $\bar {\Omega }_{\boldsymbol {s}}$, slant α_s and extension τ_s. The change of variable $\boldsymbol {v} = (m_{+}(\boldsymbol {s}))^{-1} \zeta \boldsymbol {t}^{\nu }$ leads to $\mathrm {d} \boldsymbol {t} = \nu ^{-d} \zeta ^{-d/\nu } {\prod }_{i=1}^{d} m_{i+}^{1/\nu } v_{i}^{(1-\nu )/\nu } \mathrm {d} \boldsymbol {v}$ and

$$ {\Lambda}_{\boldsymbol{s}}(A) = \nu^{-d} {\int}_{0}^{\infty} {\int}_{A} \prod\limits_{i=1}^{d} \left( m_{i+} {v}_{i}^{(1-\nu)}\right)^{1/\nu} \frac{ \phi_{d} \left( \boldsymbol{v}^{\circ} \zeta^{-1/\nu}; \bar{\Omega}_{\boldsymbol{s}} \right) {\Phi} \left( \alpha_{\boldsymbol{s}}^{\top} \boldsymbol{v}^{\circ} \zeta^{-1/\nu} + \tau_{\boldsymbol{s}} \right)} { {\Phi}\left( \tau_{\boldsymbol{s}} \{ 1 + Q_{\bar{\Omega}^{-1}_{\boldsymbol{s}}}(\alpha_{\boldsymbol{s}}) \}^{-1/2} \right) } \zeta^{-\frac{d}{\nu}-2} \mathrm{d} \boldsymbol{v} \mathrm{d} \zeta, $$

(9)

where v^∘ = (vm₊(s))^1/ν. Now, through the consecutive change of variables t = ζ^− 1/ν and $u = t Q_{\bar {\Omega }_{\boldsymbol {s}}}(\boldsymbol {v}^{\circ })^{1/2}$ we obtain

$$ \begin{array}{@{}rcl@{}} {\int}_{0}^{\infty} \phi_{d} &&\left( \boldsymbol{v}^{\circ} \zeta^{-1/\nu}; \bar{\Omega}_{\boldsymbol{s}} \right) {\Phi} \left( \alpha_{\boldsymbol{s}}^{\top} \boldsymbol{v}^{\circ} \zeta^{-1/\nu} + \tau_{\boldsymbol{s}} \right) \zeta^{-\frac{d}{\nu}-2} \mathrm{d}\zeta \\ \quad &=& \nu {\int}_{0}^{\infty} \phi_{d} \left( \boldsymbol{v}^{\circ} t ; \bar{\Omega}_{\boldsymbol{s}} \right) {\Phi} \left( \alpha_{\boldsymbol{s}}^{\top} \boldsymbol{v}^{\circ} t + \tau_{\boldsymbol{s}} \right) t^{d+\nu-1} \mathrm{d} t \\ \quad &=& (2\pi)^{-d/2} |\bar{\Omega}_{\boldsymbol{s}}|^{-1/2} \nu {\int}_{0}^{\infty} t^{d+\nu-1} \exp \left\{ -t^{2} \frac{ Q_{\bar{\Omega}_{\boldsymbol{s}}}(\boldsymbol{v}^{\circ}) }{2} \right\} {\Phi} \left( \alpha_{\boldsymbol{s}}^{\top} \boldsymbol{v}^{\circ} t + \tau_{\boldsymbol{s}} \right) \mathrm{d} t \\ \quad &=& (2\pi)^{-d/2} |\bar{\Omega}_{\boldsymbol{s}} |^{-1/2} \nu Q_{\bar{\Omega}_{\boldsymbol{s}}}(\boldsymbol{v}^{\circ})^{-(d+\nu)/2} (2\pi)^{1/2} {\int}_{0}^{\infty} u^{d+\nu-1} \phi(u) {\Phi} \left( \tilde{\alpha}_{{s}} u + \tau_{{s}} \right) \mathrm{d} u , \end{array} $$

(10)

where $\tilde {\alpha }_{\boldsymbol {s}} = \alpha _{\boldsymbol {s}}^{\top } \boldsymbol {v}^{\circ } Q_{\bar {\Omega }_{\boldsymbol {s}}}(\boldsymbol {v}^{\circ })^{-1/2} \in \mathbb R$.

The remaining integral is linked to the moments of the extended skew-normal distribution. Beranger et al. (2017, Appendix A.4) derives the result

$$ {\int}_{0}^{\infty} y^{\nu} \phi(y) {\Phi}(\alpha y + \tau) \mathrm{d} y = 2^{(\nu-2)/2} \pi^{-1/2} {\Gamma} \left( \frac{\nu+1}{2} \right) {\Psi} \left( \alpha \sqrt{\nu+1}; -\tau, \nu+1 \right), $$

and thus Eq. 10 is equal to

$$ 2^{(\nu-2)/2} \pi^{-d/2} |\bar{\Omega}|^{-1/2} \nu \left( \boldsymbol{v}^{\circ\top} \bar{\Omega}_{\boldsymbol{s}}^{-1} \boldsymbol{v}^{\circ}\right)^{-(d+1)/2} {\Gamma} \left( \frac{\nu + d}{2} \right) {\Psi} \left( \tilde{\alpha}_{\boldsymbol{s}} \sqrt{\nu+d}; -\tau_{\boldsymbol{s}}, \nu+d \right). $$

(11)

Substituting (11) into Eq. 9 completes the proof. □

Assume that $(W(\boldsymbol {t}), W(\boldsymbol {s})) \sim \mathcal {S}\mathcal {N}_{m+d}(\bar {\Omega }_{(\boldsymbol {t},\boldsymbol {s})}, \alpha _{(\boldsymbol {t},\boldsymbol {s})}, \tau _{(\boldsymbol {t},\boldsymbol {s})})$, then according to Beranger et al. (2017, Proposition 1) we have that $W(\boldsymbol {s}) \sim \mathcal {S}\mathcal {N}_{d} (\bar {\Omega }_{\boldsymbol {s}}, {\alpha }^{\ast }_{\boldsymbol {s}}, {\tau }^{\ast }_{\boldsymbol {s}})$ with

$$ \begin{array}{ccc} {\alpha}^{\ast}_{\boldsymbol{s}} = \frac{\alpha_{\boldsymbol{s}} + \bar{\Omega}_{\boldsymbol{s}}^{-1} \bar{\Omega}_{\boldsymbol{s}\boldsymbol{t}} \alpha_{\boldsymbol{t}}} {\sqrt{1 + Q_{\tilde{\Omega}^{-1}}(\alpha_{\boldsymbol{t}})}}, & {\tau}^{\ast}_{\boldsymbol{s}} = \frac{\tau_{(\boldsymbol{t},\boldsymbol{s})}}{\sqrt{1 + Q_{\tilde{\Omega}^{-1}}(\alpha_{\boldsymbol{t}})}}, & \tilde{\Omega} = \bar{\Omega}_{\boldsymbol{t}} - \bar{\Omega}_{\boldsymbol{t} \boldsymbol{s}} \bar{\Omega}_{\boldsymbol{s}}^{-1} \bar{\Omega}_{\boldsymbol{s} \boldsymbol{t}}. \end{array} $$

Additionally let u^∘ = (um₊(t))^1/ν, v^∘ = (vm₊(s))^1/ν, m₊(t) = (m₊(t₁),…, m₊(t_m)), m₊(s) = (m₊(s₁),…, m₊(s_d)), $\boldsymbol {u} \in \mathbb {R}^{m}$, $\boldsymbol {v} \in \mathbb {R}^{d}$. Noting that ${\Phi } (\tau _{(\boldsymbol {t},\boldsymbol {s})} (1+Q_{\bar {\Omega }^{-1}_{(\boldsymbol {t},\boldsymbol {s})}}(\alpha _{(\boldsymbol {t},\boldsymbol {s})}))^{-1/2})$ is equal to ${\Phi } (\tau ^{\ast }_{\boldsymbol {s}} (1+Q_{\bar {\Omega }^{-1}_{\boldsymbol {s}}}(\alpha ^{\ast }_{\boldsymbol {s}}))^{-1/2})$ and applying Lemma 2 to Eq. 6 leads to

$$ \begin{array}{@{}rcl@{}} \lambda_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}(\boldsymbol{u}){} \!&=&\!{} \pi^{-m/2} \nu^{-m} \frac{|\bar{\Omega}_{(\boldsymbol{t},\boldsymbol{s})}|^{-1/2}}{|\bar{\Omega}_{\boldsymbol{s}}|^{-1/2}} \left\{{} \frac{Q_{\bar{\Omega}_{(\boldsymbol{t},\boldsymbol{s})}} (\boldsymbol{u}^{\circ}, \boldsymbol{v}^{\circ}) } {Q_{\bar{\Omega}_{\boldsymbol{s}}} (\boldsymbol{v}^{\circ})} \right\}^{-\frac{\nu+d+m}{2}} Q_{\bar{\Omega}_{\boldsymbol{s}}} (\boldsymbol{v}^{\circ})^{-\frac{m}{2}} \frac{{\Gamma}(\frac{\nu+d+m}{2})}{{\Gamma}(\frac{\nu+d}{2})} \\ \quad &&\times \frac{ {\Psi}\left( \tilde{\alpha}_{(\boldsymbol{t}, \boldsymbol{s})} \sqrt{\nu + d + m}; -\tau_{(\boldsymbol{t}, \boldsymbol{s})}, \nu + d + m \right) } { {\Psi} \left( \tilde{\alpha}_{\boldsymbol{s}} \sqrt{\nu + d}; -\tau^{\ast}_{\boldsymbol{s}}, \nu + d \right) } \prod\limits_{i=1}^{m} \left( m_{+}(t_{i}) u_{i}^{1-\nu} \right)^{1/\nu}, \end{array} $$

where $\tilde {\alpha }_{(\boldsymbol {t}, \boldsymbol {s})} = \alpha ^{\top }_{(\boldsymbol {t}, \boldsymbol {s})} (\boldsymbol {u}^{\circ }, \boldsymbol {v}^{\circ }) Q_{\bar {\Omega }_{(\boldsymbol {t},\boldsymbol {s})}} (\boldsymbol {u}^{\circ }, \boldsymbol {v}^{\circ })^{-1/2}$ and $\tilde {\alpha }_{\boldsymbol {s}} = {\alpha }^{\ast \top }_{\boldsymbol {s}} \boldsymbol {v}^{\circ } Q_{\bar {\Omega }_{\boldsymbol {s}}} (\boldsymbol {v}^{\circ })^{-1/2}.$

Following Dombry et al. (2013) and Ribatet (2013) we can show that

$$ \frac{\lvert\bar{\Omega}_{(\boldsymbol{t},\boldsymbol{s})}\rvert}{\lvert\bar{\Omega}_{\boldsymbol{s}}\rvert} {}= {}\left\{ {}\frac{\nu + d}{ Q_{\bar{\Omega}} (\boldsymbol{v}^{\circ}) }{} \right\}^{m}{} \lvert {\Omega}_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}} \rvert, {} \frac{Q_{\bar{\Omega}_{(\boldsymbol{t},\boldsymbol{s})}} (\boldsymbol{u}^{\circ}, \boldsymbol{v}^{\circ}) } {Q_{\bar{\Omega}_{\boldsymbol{s}}} (\boldsymbol{v}^{\circ})} {}={} 1 + \frac{Q_{{\Omega}_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}} (\boldsymbol{u}^{\circ} - \mu_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}) } {\nu + d}, {\Omega}_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}} = \frac{ Q_{\bar{\Omega}_{\boldsymbol{s}}} (\boldsymbol{v}^{\circ}) }{\nu + d} \tilde{\Omega}, $$

and $\mu _{\boldsymbol {t} | \boldsymbol {s}, \boldsymbol {v}} = \bar {\Omega }_{\boldsymbol {t}\boldsymbol {s}} \bar {\Omega }_{\boldsymbol {s}}^{-1} \boldsymbol {v}^{\circ }.$ Thus we have

$$ \begin{array}{@{}rcl@{}} \lambda_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}(\boldsymbol{u}) &= \pi^{-m/2} (\nu + d)^{-m/2} \lvert{\Omega}_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}\rvert^{-1/2} \left\{ 1 + \frac{Q_{{\Omega}_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}} (\boldsymbol{u}^{\circ} - \mu_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}) } {\nu + d} \right\}^{-\frac{\nu + d + m}{2}} \\ & \quad \times \frac{{\Gamma}(\frac{\nu+d+m}{2})}{{\Gamma}(\frac{\nu+d}{2})} \frac{{\Psi} \left( \tilde{\alpha}_{(\boldsymbol{t}, {s})} \sqrt{\nu + d + m}; -\tau_{(\boldsymbol{t}, {s})}, \nu + d + m \right)} {{\Psi} \left( \tilde{\alpha}_{\boldsymbol{s}} \sqrt{\nu + d}; -\tau^{\ast}_{\boldsymbol{s}}, \nu + d \right)} \nu^{-m} \prod\limits_{i=1}^{m} \left( m_{+}(t_{i}) u_{i}^{1-\nu} \right)^{1/\nu}, \end{array} $$

and we recognise the m-dimensional Student-t density with mean μ_{t|s, v}, dispersion matrix Ω_{t|s, v} and degree of freedom ν + d.

Finally, by considering $\alpha _{\boldsymbol {t} | \boldsymbol {s}, \boldsymbol {v}} = \tilde {\omega }^{-1} \alpha _{\boldsymbol {t}}$, $\tilde {\omega } = \text {diag}(\tilde {\Omega })^{1/2},$$\tau _{\boldsymbol {t} | \boldsymbol {s}, \boldsymbol {v}} = \left (\alpha _{\boldsymbol {s}} + \bar {\Omega }_{\boldsymbol {s}}^{-1} \bar {\Omega }_{\boldsymbol {s} \boldsymbol {t}} \alpha _{\boldsymbol {t}} \right )^{\top } \boldsymbol {v}^{\circ } (d+\nu )^{1/2} Q_{\bar {\Omega }} (\boldsymbol {v}^{\circ })^{-1/2}$ and κ_{t|s, v} = −τ_{(t, s)} then it is easy to show that

$$ \begin{array}{@{}rcl@{}} {\Psi} \left( \tilde{\alpha}_{\boldsymbol{s}} \sqrt{\nu + d}; -\tau^{\ast}_{\boldsymbol{s}}, \nu + d \right) {}&=& {}{\Psi} \left( \frac{\tau_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}}{\sqrt{1+Q_{\bar{\Omega}^{-1}_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}}(\alpha_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}})}}; -\frac{\tau_{(\boldsymbol{t},\boldsymbol{s})}}{\sqrt{1+Q_{\bar{\Omega}^{-1}_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}}(\alpha_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}})}}, \nu_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}} \right) \\ &&{\Psi} \left( \tilde{\alpha}_{(\boldsymbol{t}, \boldsymbol{s})} \sqrt{\nu + d + m}; -\tau_{(\boldsymbol{t}, \boldsymbol{s})}, \right. \left. \nu + d + m \right)\\ &=& {}{\Psi} \left( {} \left( {}\alpha_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}^{\top} z + \tau_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}} \right) \sqrt{\frac{\nu_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}} + m}{\nu_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}} + Q_{\bar{\Omega}_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}}(z)}}; -\tau_{(\boldsymbol{t},\boldsymbol{s})}, \nu_{\boldsymbol{t} | \boldsymbol{s}, \boldsymbol{v}}+m \right) , \end{array} $$

where z = u^∘− μ_{t|s, v}, completes the proof. Note that $\bar {\Omega }_{\boldsymbol {t} | \boldsymbol {s}, \boldsymbol {v}}$ reduces to $\tilde {\omega }^{-1} \tilde {\Omega } \tilde {\omega }^{-1}$.

1.5 A.5 Partial derivatives of the V function of the extremal skew-t (Lemma 3)

Consider the conditional intensity function of the extremal skew-t model given in Proposition 2 with s = (s₁,…, s_m) ≡s_1:m, t = (s_m+ 1,…, s_d) ≡s_m+ 1:d, v = z_1:m and u = z_m+ 1:d. In the following the matrix notation ${\Sigma }_{a:b}= {\Sigma }_{\boldsymbol {s}_{a:b}}, {\Sigma }_{a:b;c:d}= {\Sigma }_{\boldsymbol {s}_{a:b} \boldsymbol {s}_{c:d}}$ will be used. Integration w.r.t. z_m+ 1:d gives

$$ {\Psi}_{d-m} \left( \boldsymbol{z}_{m+1:d}^{\circ}; \mu_{c}, {\Omega}_{c}, \alpha_{c}, \tau_{c}, \kappa_{c}, \nu_{c} \right) $$

(12)

where the index c represents s_m+ 1:d|s_1:m, z_1:m such that the parameters are

$$ \begin{array}{@{}rcl@{}} \mu_{c} &=& \bar{\Omega}_{m+1:d ; 1:m} \bar{\Omega}_{1:m}^{-1} \boldsymbol{z}_{1:m}^{\circ}, \quad {\Omega}_{c} = \frac{Q_{\bar{\Omega}_{1:m}} (\boldsymbol{z}_{1:m}^{\circ})}{\nu_{c}} \tilde{\Omega}_{c}, \\ \tilde{\Omega}_{c} &=& \bar{\Omega}_{m+1:d} - \bar{\Omega}_{m+1:d; 1:m} \bar{\Omega}_{1:m}^{-1} \bar{\Omega}_{1:m ; m+1:d}, \\ \tau_{c} &=& (\alpha_{1:m} + \bar{\Omega}_{1:m}^{-1} \bar{\Omega}_{1:m ; m+1:d} \alpha_{m+1:d} )^{\top} \boldsymbol{z}_{1:m}^{\circ} (\nu+m)^{1/2} Q_{\bar{\Omega}_{1:m}} (\boldsymbol{z}_{1:m}^{\circ})^{-1/2}, \\ \alpha_{c} &=& \omega_{c} \alpha_{m+1:d}, \quad \tilde{\omega}_{c} = \text{diag}(\tilde{\Omega}_{c})^{1/2}, \quad \kappa_{c} = -\tau, \quad \nu_{c} = \nu + m, \end{array} $$

with $\boldsymbol {z}_{1:m}^{\circ } = (\boldsymbol {z}_{1:m} m_{+}(\boldsymbol {s}_{1:m}))^{1/\nu }$ and $\boldsymbol {z}_{m+1:d}^{\circ } = (\boldsymbol {z}_{m+1:d} m_{+}(\boldsymbol {s}_{m+1:d}))^{1/\nu }$. According to Lemma 2, the m-dimensional marginal density is

$$ \frac{ 2^{(\nu-2)/2} \nu^{-m+1} {\Gamma} \left( \frac{m+\nu}{2} \right) {\Psi} \left( \tilde{\alpha}_{1:m} \sqrt{m+\nu}; - \tau_{1:m}^{\ast}, m+\nu \right) {\prod}_{i=1}^{m} \left( m_{i+} z_{i}^{1-\nu} \right)^{1/\nu} } { \pi^{m/2} |\bar{\Omega}_{1:m}|^{1/2} Q_{\bar{\Omega}_{1:m}} (\boldsymbol{z}_{1:m}^{\circ} )^{(m+\nu)/2} {\Phi}(\tau_{1:m}^{\ast} \{ 1 + Q_{\bar{\Omega}^{-1}_{1:m}}(\alpha^{\ast}_{1:m}) \}^{-1/2} ) }, $$

(13)

where $\tilde {\alpha }_{1:m} = \alpha _{1:m}^{\ast \top } \boldsymbol {z}_{1:m}^{\circ } Q_{\bar {\Omega }_{1:m}} (\boldsymbol {z}_{1:m}^{\circ } )^{-1/2} \in \mathbb {R}$, and m-dimensional marginal parameters

$$ \begin{array}{ll} \alpha_{1:m}^{\ast} = \frac{\alpha_{1:m} + \bar{\Omega}_{1:m}^{-1} \bar{\Omega}_{1:m; m+1:d} \alpha_{m+1:d} } {\sqrt{1+ Q_{\bar{\Omega}^{-1}_{c}} (\alpha_{m+1:d})}}, & \tau_{1:m}^{\ast} = \frac{\tau} {\sqrt{1+ Q_{\bar{\Omega}^{-1}_{c}}(\alpha_{m+1:d})}}. \end{array} $$

Combining (12) and Eq. 13 completes the proof.

Setting τ_s = 0 corresponds to an extremal skew-t model constructed from a skew-normal random field rather than an extended skew-normal field. Then

$$ m_{j+} = 2^{\nu/2} \pi^{-1/2} {\Gamma}\left( \frac{\nu + 1}{2}\right) {\Psi}(\alpha^{\ast}_{j} \sqrt{\nu +1}; \nu +1), $$

with

$$ \alpha^{\ast}_{j} = \frac{\alpha_{j} + \bar{\Omega}^{-1}_{jj} \bar{\Omega}_{j I_{j}} \alpha_{I_{j}} } {\sqrt{1 + \alpha_{I_{j}}^{\top} \left( \bar{\Omega}_{I_{j} I_{j}} - \bar{\Omega}_{I_{j} j} \bar{\Omega}^{-1}_{jj} \bar{\Omega}_{j I_{j}} \right) \alpha_{I_{j}}} }, $$

and the associated partial derivatives of the V function are equal to

$$ {\Psi}_{d-m} \left( \boldsymbol{z}_{m+1:d}^{\circ}; \mu_{c}, {\Omega}_{c}, \alpha_{c}, \tau_{c}, 0, \nu_{c} \right) \frac{ 2^{\nu/2} {\Gamma} \left( \frac{m+\nu}{2} \right) {\Psi} \left( \tilde{\alpha}_{1:m} \sqrt{m+\nu}; m+\nu \right) {\prod}_{i=1}^{m} \left( m_{i+} z_{i}^{1-\nu} \right)^{1/\nu} } { \pi^{m/2} \nu^{m-1} |\bar{\Omega}_{1:m}|^{1/2} Q_{\bar{\Omega}_{1:m}} (\boldsymbol{z}_{1:m}^{\circ} )^{(m+\nu)/2} }, $$

with parameters defined as in Lemma 3.

Setting α_s = α_1:d = 0 and τ_s = 0 leads to the extremal-t model for which

$$ m_{i+} = 2^{(\nu-2)/2} \pi^{-1/2} {\Gamma}\left( \frac{\nu + 1}{2}\right) \equiv m_{+}, $$

and the partial derivatives of the V function are

$$ {\Psi}_{d-m} \left( \boldsymbol{z}_{m+1:d}^{1/\nu}; \mu^{\prime}_{c}, {\Omega}^{\prime}_{c}, \nu_{c} \right) \frac{ 2^{(\nu-2)/2} \nu^{-m+1} {\Gamma} \left( \frac{m+\nu}{2} \right) {\prod}_{i=1}^{m} z_{i}^{(1-\nu)/\nu} } { \pi^{m/2} |\bar{\Omega}_{1:m}|^{1/2} Q_{\bar{\Omega}_{1:m}} (\boldsymbol{z}_{1:m}^{1/ \nu} )^{(m+\nu)/2} m_{+} }, $$

where $\mu ^{\prime }_{c} = \bar {\Omega }_{m+1:d ; 1:m} \bar {\Omega }_{1:m}^{-1} \boldsymbol {z}_{1:m}^{1/ \nu }$ and ${\Omega }^{\prime }_{c} = \frac {Q_{\bar {\Omega }_{1:m}} (\boldsymbol {z}_{1:m}^{1/\nu })}{\nu _{c}} \bar {\Omega }_{c}$.

Appendix B: Simulation Tables

Table 4 Absolute biases $|\bar {\hat {\theta }}_j - \theta _j|$ for $\hat {\theta }_j = (\hat {\eta }_j, \hat {r}_j)$ and $\hat {\theta }_j = (\hat {\eta }_j, \hat {r}_j, \hat {\beta }_{1j}, \hat {\beta }_{2j})$ the parameter vectors of the extremal-t and extremal skew-t models, using the full likelihood Type I and Type II approximations given in Table 1 when d = 20,50 and 100 sites are considered. Calculations are based on 500 replicate maximisations

Table 5 RMSEs for $\hat {\theta }_j = (\hat {\eta }_j, \hat {r}_j)$ and $\hat {\theta }_j = (\hat {\eta }_j, \hat {r}_j, \hat {\beta }_{1j}, \hat {\beta }_{2j})$ the parameter vectors of the extremal-t and extremal skew-t models using the j-wise composite likelihood when v = 1 and d = 20. The case j = d corresponds to full likelihood estimation using approximation Type II from Table 1 . Calculations are based on 500 replicate maximisations

Table 6 Absolute biases $|\bar {\hat {\theta }}_j - \theta _j|$ for $\hat {\theta }_j = (\hat {\eta }_j, \hat {r}_j)$ and $\hat {\theta }_j = (\hat {\eta }_j, \hat {r}_j, \hat {\beta }_{1j}, \hat {\beta }_{2j})$ the parameter vectors of the extremal-t and extremal skew-t models using the j-wise composite likelihood when v = 1 and d = 20. The case j = d corresponds to full likelihood estimation using approximation Type II from Table 1. Calculations are based on 500 replicate maximisations.

Appendix C: Exact simulation of Extremal skew-t Max Stable Process with Hitting Scenarios

Below we provide pseudo-code for exact simulation of extremal skew-t max stable processes with unit Fréchet marginal distributions using Algorithm 2 of Dombry et al. (2016), extended to include the hitting scenario in the output. This requires the simulation of an extended skew-t distribution; here we use rejection sampling and the stochastic representation given in Arellano-Valle and Genton (2010). The simpler extremal-t max stable process only requires the simulation of a multivariate t-distribution and therefore does not use rejection sampling; this simpler algorithm is also given below.

When simulating N independent replicates for d sites with the Dombry et al. (2016) algorithm, it is much more efficient to have the sites in the outer loop and the replicates in the inner loop, because derivations of quantities from the distribution of $(W(s) / W(s_{0}))_{s \in \mathcal {S}}$ are then only performed once for each site (lines 3 to 7 in the skew-t code), irrespective of the number of replicates required. In practice these quantities should be calculated on the log scale to avoid numerical issues.

In the algorithm below, the input Σ_d is derived from the correlation function ρ(h). The normalization in line 6 is needed for the simulation of an extended skew-t distribution. Matrix multiplication is not needed here because ω is a diagonal matrix. The term Exp(1) refers to a standard exponential distribution, $t_{\nu _{d}}$ is a univariate t-distribution with ν_d degrees of freedom, N(0, 1) is a standard univariate normal distribution, and $\chi ^{2}_{\nu _{d}}$ is a chi-squared distribution with ν_d degrees of freedom. The function Ψ(⋅; ν_d) is the distribution function of a univariate t-distribution, as used in Eq. (A.5) The code in line 16 simulates from a multivariate t-distribution with shape matrix $\hat {\bar {\Sigma }}_{d}$ and ν_d degrees of freedom. Lines 20 and 24 are identical by intent.

The index j in the code corresponds to the s₀ site. We recommend the use of the eigendecomposition, which is more stable than the Cholesky decomposition. Moreover, $\bar {\Sigma }^{}_{d}$ is positive semi-definite as the j th row and columns are zero by construction. If the Cholesky decomposition were used then the code would need to handle the singular component explicitly. The eigendecomposition is slower, but it can be evaluated outside the loop over the observations (in line 7) and therefore only d decompositions are required for any N.

The do-while loop in line 12 is the rejection sampling needed to simulate from a multivariate extended skew-t distribution. The Dombry et al. (2016) algorithm also has a rejection step, with B = 0 in the code indicating rejection via exceeding an observation on an already simulated site (i.e. on a site with index less than j). If the simulation is not rejected (line 22) then the outputs are set. A simulated process will always update the value on the j th site, because there is a singular component X[j] ≡ 1 and therefore the code would otherwise not enter the while loop at line 10. If the code enters the while loop (line 10), it breaks out of it when $\tilde {E}$ is small enough that the j th site simulation can never exceed the existing value. The vector V counts the number of times the while loop executes for each replicate. This ultimately provides the hitting scenario H.