2.3.1. Coefficients $ \tau_C(\delta)$

Simpson et al. [Reference Simpson, Wadsworth and Tawn42] sought to examine the extremal dependence structure of a random vector through determination of the cones $ \mathbb{E}_C$ for which $ \nu\big(\mathbb{E}_C\big)>0$ . Direct consideration of (hidden) regular variation conditions on these cones is impeded by the fact that $ {\mathbb P}({\boldsymbol{X}}_P/b(t) \in B) = 0$ for all $ B \subset \mathbb{E}_C$ , $ C \neq D$ , since no components of $ {\boldsymbol{X}}_P/b(t)$ are equal to zero for $ t< \infty$ . Simpson et al. [Reference Simpson, Wadsworth and Tawn42] circumvent this issue by assuming that if $ \nu\big(\mathbb{E}_C\big)>0$ , then there exists $ \delta<1$ such that

(2.5) \begin{align}\lim_{t \to \infty} &t{\mathbb P}\Big(\!\min_{i \in C}X_{P,i}> xt, \max_{j \in D \setminus C}X_{P,j} \leq y t^\delta\Big) \nonumber\\&= \lim_{t \to \infty} t{\mathbb P}\Big(\!\min_{i \in C}X_{P,i}/t> x, \max_{j \in D \setminus C}X_{P,j}/t \leq y t^{\delta-1}\Big)>0, \qquad x,y>0. \end{align}

Consequently, under normalization by t, components of the random vector indexed by C remain positive, whereas those indexed by $ D\setminus C$ concentrate at zero. Note that if the assumption (2.5) holds for some $ \delta<1$ , then it also holds for all $ \delta^{\prime} \in [\delta,1]$ . Simpson et al. [Reference Simpson, Wadsworth and Tawn42] expanded the assumption (2.5) to

(2.6) \begin{align}{\mathbb P}\Big(\!\min_{i \in C}X_{P,i}> xt, \max_{j \in D \setminus C}X_{P,j} \leq y t^\delta\Big) \in \mathrm{RV}_{-1/\tau_C(\delta)}^{\infty}, \qquad \delta \in [0,1] ,\end{align}

where (2.6) is viewed as a function of t, and the regular variation coefficients $ \tau_C(\delta) \in (0,1]$ . For a fixed $ \delta$ , $ \tau_C(\delta)<1$ implies either that $ \nu\big(\mathbb{E}_C\big)=0$ , or that $ \nu\big(\mathbb{E}_C\big)>0$ , but that $ \delta$ is too small for (2.5) to hold; see [Reference Simpson, Wadsworth and Tawn42] for further details. Considering the coefficients $ \tau_C(\delta)$ over all C and $ \delta \in [0,1]$ provides information about the cones on which $ \nu$ concentrates.

2.3.2. Angular dependence function $ \lambda(\boldsymbol{\omega})$

Wadsworth and Tawn [Reference Wadsworth and Tawn44] detail a representation for the tail of $ {\boldsymbol{X}}_P$ where the scaling functions are of different order in each component. They focus principally on a sequence of univariate regular variation conditions, characterizing

(2.7) \begin{align}{\mathbb P}\big({\boldsymbol{X}}_P>t^{\boldsymbol{\omega}}\big) = \ell(t;\, \boldsymbol{\omega})t^{-\lambda(\boldsymbol{\omega})}, \qquad \boldsymbol{\omega} \in \mathcal{S}_{\Sigma}=\left\{\boldsymbol{\omega} \in [0,1]^d \,:\, \sum_{j=1}^d \omega_j = 1\right\}, \end{align}

where $ \ell(t;\, \boldsymbol{\omega}) \in \mathrm{RV}_0^\infty$ for each $ \boldsymbol{\omega}$ and $ \lambda\,:\,\mathcal{S}_{\Sigma} \to [0,1]$ . Equivalently, $ {\mathbb P}\big({\boldsymbol{X}}_E > {\boldsymbol{\omega}}v\big) = \ell\big(e^v;\, \boldsymbol{\omega}\big)e^{-\lambda(\boldsymbol{\omega})v}$ . When all components of $ \boldsymbol{\omega}$ are equal to $ 1/d$ , connection with hidden regular variation on the cone $ \mathbb{E}_D$ is restored, and we have $ \eta_D = d\lambda(1/d,\ldots,1/d)$ . When the subcone $ \mathbb{E}_D$ of $ \mathbb{E}$ is charged with mass in the limit (2.1), we have $ \lambda(\boldsymbol{\omega}) = \max_{1\leq j \leq d}\omega_j$ . One can equally focus on subvectors indexed by C to define $ \lambda_{C}(\boldsymbol{\omega})$ for $ \boldsymbol{\omega}$ in a $ (|C|-1)$ -dimensional simplex; we continue to have $ \eta_C = |C|\lambda_C(1/|C|,\ldots,1/|C|)$ and $ \nu\big(\mathbb{E}_C\big)>0$ implies $ \lambda_{C}(\boldsymbol{\omega}) = \max_{1 \leq j \leq |C|} \omega_j$ .

2.5.1. Background

Let $ {\boldsymbol{X}}_1,\ldots,{\boldsymbol{X}}_n$ be independent and identically distributed random vectors in $ {\mathbb R}^d$ . A random set $ N_n=\big\{{\boldsymbol{X}}_1/r_n,\ldots,{\boldsymbol{X}}_n/r_n\big\}$ represents a scaled n-point sample cloud. We consider situations in which there exists a scaling sequence $ r_n>0$ , $ r_n\to\infty$ , such that scaled sample clouds $ N_n$ converge onto a deterministic set containing at least two points. Figure 1 illustrates examples of sample clouds for which a limit set exists. Let $ {{\mathcal{K}}}_d$ denote the family of non-empty compact subsets of $ {\mathbb R}^d$ , and let $ d_H({\cdot},{\cdot})$ denote the Hausdorff distance between two sets [Reference Matheron35]. A sequence of random sets $ N_n$ in $ {{\mathcal{K}}}_d$ converges in probability onto a limit set $ G\in{{\mathcal{K}}}_d$ if $ d_H(N_n,G)\buildrel{\mathbb P}\over\rightarrow 0$ for $ n\to\infty$ . The following result gives convenient criteria for showing convergence in probability onto a limit set; see [Reference Balkema, Embrechts and Nolde2].

Figure 1.

Sample clouds of $ n=10^5$ points simulated from meta-Gaussian distributions with standard exponential margins and copula correlation parameter $ \rho=0.5$ (left panel), $ \rho=0$ (middle panel), and $ \rho=-0.5$ (right panel). The samples are scaled by the factor $ r_n=\log n$ . See Examples 4.1.1 and 4.1.2 for details.

Limit sets under various assumptions on the underlying distribution have been derived in [Reference Geffroy20, Reference Geffroy21, Reference Fisher18, Reference Davis, Mulrow and Resnick12, Reference Balkema, Embrechts and Nolde2]. Kinoshita and Resnick [Reference Kinoshita and Resnick30] give a complete characterization of the possible limit sets, as well as describing the class of distribution functions for which sample clouds can be scaled to converge (almost surely) onto a limit set. Furthermore, convergence in probability onto a limit set is implied by the tail large deviation principle studied in [Reference De Valk15, Reference De Valk14].

Kinoshita and Resnick [Reference Kinoshita and Resnick30] showed that if sample clouds can be scaled to converge onto a limit set almost surely, then the limit set is compact and star-shaped. A set G in $ {\mathbb R}^d$ is star-shaped if $ \boldsymbol{x}\in G$ implies $ t\boldsymbol{x} \in G$ for all $ t \in [0,1]$ . For a set $ G\in{{\mathcal{K}}}_d$ , if the line segment $ {\textbf{0}}+t \boldsymbol{x}$ , $ t\in [0,1)$ , is contained in the interior of G for every $ \boldsymbol{x}\in G$ , then G can be characterized by a continuous gauge function:

\begin{equation*}g(\boldsymbol{x})=\inf\{t\ge0\,:\,\ \boldsymbol{x}\in tG\},\qquad \boldsymbol{x}\in{\mathbb R}^d.\end{equation*}

A gauge function satisfies homogeneity— $ g(t\boldsymbol{x})=tg(\boldsymbol{x})$ for all $ t>0$ —and the set G can be recovered from its gauge function via $ G=\big\{\boldsymbol{x}\in{\mathbb R}^d\,:\,\ g(\boldsymbol{x})\le1\big\}.$ Examples of a gauge function include a norm $ \|{\cdot}\|$ on $ {\mathbb R}^d$ , in which case $ G=\{\boldsymbol{x}\in{\mathbb R}^d\,:\,\ \|\boldsymbol{x}\|\le1\}$ is the unit ball in that norm.

The shape of the limit set conveys information about extremal dependence properties of the underlying distribution. In particular, [Reference Balkema and Nolde4] make a connection between the shape of the limit set and asymptotic independence, whilst [Reference Nolde36] links its shape to the coefficient of residual tail dependence. We emphasize that the shape of the limit set depends on the choice of marginal distributions, as well as the dependence structure. For example, if the components of $ (X_1,X_2)$ are independent with common marginal distribution, then $ G = \big\{(x,y) \in \mathbb{R}_+^2\,:\, x+y \leq 1\big\}$ if the margins are exponential; $ G = \big\{(x,y) \in \mathbb{R}^2\,:\, |x|+|y| \leq 1\big\}$ if the margins are Laplace; and $ G = \big\{(x,y) \in \mathbb{R}_+^2\,:\, \big(x^\beta +y^\beta\big)^{1/\beta} \leq 1\big\}$ if the margins are Weibull with shape $ \beta>0$ . In contrast, if the margins are exponential but G takes the latter form, this implies some dependence between the components.

2.5.2. Conditions for convergence onto a limit set

Proposition 2.1 provides necessary and sufficient conditions for convergence onto the limit set G, but these conditions are not particularly helpful for determining the form of G in practice.

In the following proposition, we state a criterion in terms of the joint probability density for convergence of suitably scaled random samples onto a limit set. This result is an adaptation of Proposition 3.7 in [Reference Balkema and Nolde4]. The marginal tails of the underlying distribution are assumed to be asymptotically equal to a von Mises function. A function of the form $ e^{-\psi}$ is said to be a von Mises function if $ \psi$ is a $ C^2$ function with a positive derivative such that $ (1/\psi^{\prime}(x))^{\prime}\to 0$ for $ x\to\infty$ . This condition on the margins says that they are light-tailed and lie in the maximum domain of attraction of the Gumbel distribution; i.e., for a random sample from such a univariate distribution, coordinatewise maxima can be normalized to converge weakly to the Gumbel distribution (see [Reference Resnick38, Proposition 1.1]).

Proposition 2.2. Let the random vector $ {\boldsymbol{X}}$ on $ [0,\infty)^d$ have marginal distribution functions asymptotically equal to a von Mises function, i.e. $ 1-F_j(x) \sim e^{-\psi_j(x)}$ for $ \psi_j(x)\sim \psi(x)$ , $ x\to\infty$ ( $ j=1,\ldots,d$ ), and a joint probability density f satisfying

(2.10) \begin{equation}\dfrac{{-}\log f(t\boldsymbol{x}_t)}{\psi(t)}\to g_*(\boldsymbol{x}),\qquad t\to\infty,\qquad \boldsymbol{x}_t\to\boldsymbol{x},\qquad \boldsymbol{x}\in[0,\infty)^d\end{equation}

for a continuous function $ g_*$ on $ [0,\infty)^d$ , which is positive outside a bounded set. Then a sequence of scaled random samples $ N_n=\{{\boldsymbol{X}}_1/r_n,\ldots,{\boldsymbol{X}}_n/r_n\}$ from f converges in probability onto a limit set G with $ G=\{\boldsymbol{x}\in[0,\infty)^d \,:\, g_*(\boldsymbol{x})\le 1\}$ . The scaling sequence $ r_n$ can be chosen as $ \psi(r_n)\sim \log n$ . Moreover, $ \max G=(1,\ldots,1)$ .

Proof. The mean measure of $ N_n$ is given by $ n{\mathbb P}({\boldsymbol{X}}/r_n\in{\cdot})$ with intensity $ h_n(\boldsymbol{x}) = nr_n^df(r_n\boldsymbol{x})$ . We show the convergence of that mean measure onto G, implying convergence of scaled samples $ N_n$ ; see [Reference Balkema, Embrechts and Nolde2, Proposition 2.3]. By (2.10) and the choice of $ r_n$ , we have

(2.11) \begin{equation}{-}\log f(r_n\boldsymbol{x}_n)/\log n\sim {-}\log f(r_n\boldsymbol{x}_n)/\psi(r_n)\to g_*(\boldsymbol{x}), \quad n\to\infty,\quad \boldsymbol{x}_n\to\boldsymbol{x}.\end{equation}

Continuous convergence in (2.11) with $ g_*$ continuous implies uniform convergence on compact sets. Hence, $ g_*$ is bounded on compact sets. For $ G=\{g_*(\boldsymbol{x})\le1\}$ , we have $ g_*(\boldsymbol{x})<1$ on the interior of G and $ g_*(\boldsymbol{x})>1$ on the complement of G. Furthermore, applying L’Hôpital’s rule and Lemma 1.2(a) in [Reference Resnick38], we have

\begin{equation*}\log r_n/\psi(r_n)\sim (1/\psi^{\prime}(r_n))/r_n\to 0,\qquad r_n\to\infty.\end{equation*}

Combining these results, we see that $ {-}\log h_n(\boldsymbol{x}_n)\sim (g_*(\boldsymbol{x}_n)-1)\log n$ , which diverges to $ -\infty$ on the interior of G and to $ +\infty$ outside of G. This implies that

\begin{equation*}h_n(\boldsymbol{x}_n)\to\begin{cases} \infty, & \boldsymbol{x}\in G^o,\\ 0, & \boldsymbol{x}\in G^c,\end{cases} \end{equation*}

giving convergence (in probability) of $ N_n$ onto the limit set G.

The form of the margins $ 1-F_j(x) \sim e^{-\psi_j(x)}$ with $ \psi_j(x) \sim \psi(x)\to\infty$ gives $ {-}\log(1-F_j(x))\sim\psi(x)$ ; i.e.,

\begin{equation*}{-}\log\big(1-F_j(r_n)\big)\sim \psi(r_n)\sim\log n,\qquad n\to\infty.\end{equation*}

This choice of $ r_n$ implies that the coordinatewise maxima scaled by $ r_n$ converge in probability to 1 [Reference De Haan13, Reference Gnedenko22], so that $ \max G = (1,\ldots,1).$

When the margins are standard exponential, $ \psi(t)=t$ . Hence, for the random vector $ {\boldsymbol{X}}_E$ with a Lebesgue density $ f_E$ on $ \mathbb{R}^d_+$ , the condition (2.10) is equivalent to

(2.12) \begin{align}{-}\log f_E(t\boldsymbol{x}_t)/t\to g_*(\boldsymbol{x}), \qquad t \to \infty,\qquad \boldsymbol{x}_t\to\boldsymbol{x},\qquad \boldsymbol{x} \in [0,\infty)^d,\end{align}

with the limit function $ g_*$ equal to the gauge function g.

Whilst the assumption of a Lebesgue density might appear strict, it is a common feature in statistical practice of extreme value analysis. The assumption permits simple elucidation of the connection between different representations for multivariate extremes. Furthermore, many statistical models, including elliptical distributions and vine copulas [Reference Bedford and Cooke7, Reference Bedford and Cooke8, Reference Joe28], are specified most readily in terms of their densities.

Convergence at the density level such as in (2.10) may not always hold. The condition requires the limit function and hence the gauge function of the limit set to be continuous, excluding limit sets for which rays from the origin cross the boundary in more than one point. We provide an example of such a situation in Section 4; see Example 4.1.2. A less restrictive set of sufficient conditions for convergence of sample clouds onto a limit set can be obtained using the survival function. The following proposition is Theorem 2.1 in [Reference Davis, Mulrow and Resnick12], with a minor reformulation in terms of scaling.

Proposition 2.3. Suppose that the random vector $ {\boldsymbol{X}}$ has support on $ [0,\infty)^d$ ; the margins are asymptotically equal to a von Mises function, $ 1-F_j(x) \sim e^{-\psi(x)}$ for $ x\to\infty$ ( $ j=1,\ldots,d$ ); and the joint survival function satisfies

(2.13) \begin{equation}\dfrac{{-}\log {\mathbb P}({\boldsymbol{X}} \geq t\boldsymbol{x})}{\psi(t)}\to g_*(\boldsymbol{x}),\qquad t\to\infty,\qquad \boldsymbol{x}\in [0,\infty)^d \setminus\{0\}.\end{equation}

Further assume that $ g_*$ is strictly increasing, so that $ g_*(\boldsymbol{x})<g_*({\boldsymbol{y}})$ if $ \boldsymbol{x} \leq {\boldsymbol{y}}$ and $ \boldsymbol{x} \neq {\boldsymbol{y}}$ . Then for $ r_n$ satisfying $ \psi(r_n) \sim \log n$ , the sample cloud $ N_n=\{{\boldsymbol{X}}_1/r_n,\ldots,{\boldsymbol{X}}_n/r_n\}$ converges onto $ G=\big\{\boldsymbol{x}\in[0,\infty)^d \,:\, g_*(\boldsymbol{x})\le 1\big\}$ .

2.5.3. Marginalization

When $ d>2$ , a key question is the marginalization from dimension d to dimension $ m<d$ . We prove below that, as long as the minimum over each coordinate of g is well-defined, the gauge function determining the limit set in m dimensions is found through minimizing over the coordinates to be marginalized.

A continuous map h from the vector space V into the vector space $ \widetilde V$ is positive-homogeneous if $ h(r\boldsymbol{x})=rh(\boldsymbol{x})$ for all $ \boldsymbol{x}\in V$ and all $ r>0$ . If $ \widetilde V={\mathbb R}^{m}$ , the map h is determined by the m coordinate maps $ h_j\,:\,V\to{\mathbb R}$ , $ j=1,\ldots,m$ , and in this case it suffices that these maps are continuous and positive-homogeneous.

Convergence onto a limit set is preserved under linear transformations (e.g. Lemma 4.1 in [Reference Nolde36]) and more generally under continuous positive-homogeneous maps with the same scaling sequences (Theorem 1.9 in [Reference Balkema and Nolde6]). A consequence of the latter result, referred to as the Mapping Theorem, is that projections of sample clouds onto lower-dimensional subspaces also converge onto a limit set.

Proposition 2.4. Let $ N_n$ be an n-point sample cloud from a distribution of the random vector $ {\boldsymbol{X}}$ on $ {\mathbb R}^d$ . Assume $ N_n$ converges in probability, as $ n\to\infty$ , onto a limit set $ G=\{\boldsymbol{x}\in{\mathbb R}^d \,:\, g(\boldsymbol{x})\le1\}$ for a gauge function g. Let $ \widetilde{\boldsymbol{X}} = (X_i)_{i\in I_m}$ denote an m-dimensional marginal of $ {\boldsymbol{X}}$ , where $ I_m\subset I=\{1,\ldots,d\}$ is an index set with $ |I_m|=m$ . Sample clouds from $ \widetilde{\boldsymbol{X}}$ also converge, with the same scaling, and the limit set $ \tilde G = P_m(G) = \big\{{\boldsymbol{y}}\in{\mathbb R}^m \,:\, \tilde g({\boldsymbol{y}})\le1\big\}$ , where $ P_m$ is a projection map onto the coordinates of $ \widetilde{\boldsymbol{X}}$ and

\begin{equation*}\tilde g({\boldsymbol{y}}) = \min_{\{x_i \,:\, i\in I\setminus I_m\}} g(\boldsymbol{x}),\qquad \boldsymbol{x}=(x_1,\ldots,x_d),\qquad {\boldsymbol{y}}=(x_i)_{i\in I_m}.\end{equation*}

An illustration of this result is given in Section 4.2.

3.3.1. Connections to limit set G

In two dimensions, the coefficients $ \tau_{1}(\delta)$ and $ \tau_{2}(\delta)$ provide a somewhat complementary concept to the function $ \lambda(\boldsymbol{\omega})$ . Rather than considering the impact of the limit set G on the shape of the function defined by both variables exceeding thresholds growing at different rates, we are considering what is occurring when one variable exceeds a growing threshold and the other is bounded above by a certain lesser growth rate. The left and centre panels in Figure 4 provide an illustration of $ \lambda(\boldsymbol{\omega})$ and $ \tau_j(\delta)$ in two dimensions.

Figure 4.

Illustration of $ \lambda(\boldsymbol{\omega})$ and $ \tau_{j}(\delta)$ for the gauge function $ g(x,y) = \max\{(x-y)/\theta,(y-x)/\theta,(x+y)/(2-\theta)\}$ . In the left and centre panels the unit level set of g is illustrated with black dashed lines. The left panel illustrates $ \lambda(\omega,1-\omega)$ with blue solid lines. The centre panel illustrates $ \tau_j(\delta)$ with purple solid lines: $ \tau_1(\delta), \delta \in [0,1]$ is represented by the values below the main diagonal, whilst $ \tau_2(\delta)$ is represented by the values above. The set G is added on both panels with dashed lines. The right panel illustrates $ \tau_1(\delta)$ and $ \tau_2(\delta)$ in terms of the gauge function.

Define the region

\begin{equation*}R_{C,\delta} = (1,\infty]^C \times [0,\delta]^{D \setminus C} = \{\boldsymbol{x}\,:\, x_i \in (1,\infty], i \in C, x_j \in [0,\delta], j \in D\setminus C\},\end{equation*}

so that, for example, when $ d=3$ , $ R_{\{1,3\},\delta} = (1,\infty] \times [0,\delta] \times (1,\infty]$ .

Proposition 3.5. Suppose that the sample cloud $ N_n = \big\{{\boldsymbol{X}}_E^1/\log n,\ldots,{\boldsymbol{X}}_E^n/\log n\big\}$ converges onto a limit set G, and that the assumption in (2.6) holds. For $ \delta \in [0,1]$ and $ C \subset D$ ,

\begin{align*}\tau_C(\delta) &= r_{C,\delta} = \min\left\{r \in [0,1] \,:\, r R_{C,\delta} \cap G =\emptyset \right\}.\end{align*}

The coefficient $ \tau_{D} = \eta_{D}$ , and does not depend on $ \delta$ .

Proof. The coefficient $ \tau_{D}$ describes the order of hidden regular variation on the cone $ (0,\infty]^d$ , which is precisely the same as $ \eta_{D}$ . For $ \tau_C(\delta)$ , $ |C|<d$ , we consider the function of t given by

\begin{align*}{\mathbb P}\Big(\!\min_{i \in C} X_{P,i}\gt tx, \max_{j \in D \setminus C} X_{P,j} \leq y t^\delta\Big) \in \mathrm{RV}_{-1/\tau_C(\delta)}^\infty, \qquad 0\lt x,y\lt \infty.\end{align*}

Take $ x=y=1$ . Then

(3.6) \begin{align}\tau_C(\delta) &= \lim_{t \to \infty} \frac{{-}\log{\mathbb P}(X_{P,1}>t)}{{-}\log {\mathbb P}\big(\!\min_{i \in C} X_{P,i}>t, \max_{j \in D \setminus C} X_{P,j} \leq t^\delta\big)} \notag \\[5pt]&= \lim_{t \to \infty} \frac{{-}\log{\mathbb P}(X_{E,1}>t)}{{-}\log{\mathbb P}\big(\!\min_{i \in C} X_{E,i}>t, \max_{j \in D \setminus C} X_{E,j} \leq \delta t\big)} ,\end{align}

where the denominator in (3.6) can be expressed as $ {-}\log{\mathbb P}({\boldsymbol{X}}_E \in t R_{\delta})$ . As in the proof of Proposition 3.3, the convergence onto the limit set and exponential margins enable us to conclude that $ {-}\log {\mathbb P}({\boldsymbol{X}}_E \in t r_{C,\delta} R_{\delta}) \sim t$ , and hence $ {-}\log {\mathbb P}({\boldsymbol{X}}_E \in t R_{\delta}) \sim tr_{C,\delta}^{-1}$ . Combining with (3.6) gives $ \tau_C(\delta) = r_{C,\delta}$ .

In the two-dimensional case, it is possible to express $ \tau_j(\delta)$ simply in terms of the gauge function. For higher dimensions, we refer to Proposition 3.7.

The right panel of Figure 4 provides an illustration: in blue (bottom right), the value of $ \delta$ is such that $ \tau_1(\delta)<1$ ; in red (top left), the value of $ \delta$ is such that $ \tau_2(\delta)=1$ . Further detail on this example is given in Section 4.1.5.

The question arises: does Proposition 3.6 still hold for $ d>2$ , $ |C| = 1$ ? Let $ g_C$ denote the gauge function for the limit set of $ (X_{E,j}\,:\, j \in C)$ . By Proposition 2.4, we know that

\begin{equation*}g_{i,j}(x_i,x_j) = \min_{\boldsymbol{x}_{-i,-j} \in [0,\infty)^{d-2}}g_D(\boldsymbol{x}).\end{equation*}

Therefore, equality will hold if

\begin{equation*}\arg\min_{\boldsymbol{x}_{-i,-j} \in [0,\infty)^{d-2}}g_D(\boldsymbol{x}) \in [0,\delta]^{d-2}.\end{equation*}

Note that the dimension does indeed play a key role here: when looking at $ \tau_j(\delta)$ for a d-dimensional problem, we are looking at the situation where $ d-1$ coordinates are bounded above by a growth rate determined by $ \delta$ . In contrast, if we marginalize and look at $ \tau_j(\delta)$ for a two-dimensional problem, the $ d-2$ coordinates that we have marginalized over are unrestricted and so can represent small or large values. Thus, the answer to our question is negative in general.

Proposition 3.7 details the precise value of $ \tau_C(\delta)$ in terms of g for any dimension d. In a similar spirit to that of Section 3.2, define the boundary of the region $ R_{C,\delta}$ as

\begin{equation*}B_{C,\delta} = B_{C,\delta}^1 \cup B_{C,\delta}^\delta,\end{equation*}

where

\begin{align*}B_{C,\delta}^1 &= \bigcup_{i \in C} \big\{\boldsymbol{x}\in\mathbb{R}^d_+\,:\, x_i = 1, x_j \geq 1 \forall j \in C \setminus i, x_k \leq \delta \forall k \in D \setminus C \big\},\\B_{C,\delta}^\delta &= \bigcup_{i \in D \setminus C} \big\{\boldsymbol{x}\in\mathbb{R}^d_+\,:\, x_i = \delta, x_j \leq \delta \forall j \in (D\setminus C) \setminus i, x_k \geq 1 \forall k \in C \big\},\end{align*}

so, for example, when $ d=3$ ,

\begin{align*}B_{\{1,3\},\delta} = \, &\big\{\boldsymbol{x}\in \mathbb{R}^3_+\,:\, x_1 = 1, x_2 \leq \delta, x_3 \geq 1\big\}\\&\cup \big\{\boldsymbol{x}\in \mathbb{R}^3_+\,:\, x_1 \geq 1, x_2 = \delta, x_3 \geq 1\big\} \cup \big\{\boldsymbol{x}\in \mathbb{R}^3_+\,:\, x_1 \geq 1, x_2 \leq \delta, x_3 = 1\big\}.\end{align*}

For $ C=D$ , $ R_D = (1,\infty]^d$ , and $ B_D =\{\boldsymbol{x}\,:\, \min\!(\boldsymbol{x}) = 1\}$ .

Proposition 3.7. Assume the conditions of Proposition 3.5. For any $ C \subseteq D$ ,

\begin{equation*}\tau_C(\delta) = \left[\!\min_{\boldsymbol{y} \in B_{C,\delta}}g(\boldsymbol{y})\right]^{-1} =\left[\!\min_{\boldsymbol{y} \in B_{C,\delta}^1}g(\boldsymbol{y})\right]^{-1}. \end{equation*}

When $ d=2$ and $ |C|=1$ , we note that $ B_{\{j\},\delta}^1 = \{\boldsymbol{x}\,:\,x_j = 1, x_i \leq \delta\}$ , which gives the equality in Proposition 3.6.

Proof of Proposition 3.4. The proof follows exactly as for the first equality in Proposition 3.7, replacing $ R_{C,\delta}, B_{C,\delta}$ , and $ r_{C,\delta}$ with $ R_{\boldsymbol{\omega}}, B_{\boldsymbol{\omega}}$ , and $ r_{ \boldsymbol{\omega}}$ .

3.3.2. Estimation of coefficients $ \tau_C(\delta)$

When $ C=D$ , Equation (2.6) yields

\begin{equation*}{\mathbb P}\Big(\!\min_{i \in D} X_{P,i}> t\Big) \in \mathrm{RV}^{\infty}_{-1/\tau_{D}},\end{equation*}

implying that $ \tau_D$ can be estimated as the reciprocal of the tail index of the so-called structure variable $ \min_{i \in D} X_{P,i}$ . This is identical to estimating the residual tail dependence coefficient $ \eta_D$ , for which the Hill estimator is commonly employed. However, for C with $ |C|<d$ , we assume

\begin{equation*}{\mathbb P}\left(\!\min_{i \in C} X_{P,i}> t , \max_{j \in D \setminus C} X_{P,j}< t^{\delta} \right) \in \mathrm{RV}^{\infty}_{-1/\tau_{C}(\delta)},\end{equation*}

but this representation does not lend itself immediately to an estimation strategy, as there is no longer a simple structure variable for which $ 1/\tau_{C}(\delta)$ is the tail index.

In order to allow estimation, [Reference Simpson, Wadsworth and Tawn42] considered

\begin{equation*}{\mathbb P} \left(\!\min_{i \in C} X_{P,i}> t , \max_{j \in D \setminus C} X_{P,j}< \left(\!\min_{i \in C} X_{P,i} \right)^{\delta}\right),\end{equation*}

but they offered only empirical evidence that the assumed index of regular variation for this probability was the same as in (2.6). We now prove this to be the case.

Define

\begin{equation*}R_{C,\delta}^x = \Big\{\boldsymbol{x}\in \mathbb{R}^d_+\,:\, x_i>1, i \in C,x_j \leq \delta \min_{l \in C} x_l, j \in D\setminus C\Big\},\end{equation*}

and let $ B_{C,\delta}^x = B_{C,\delta}^{x,1} \cup B_{C,\delta}^{x,\delta}$ denote its boundary, where

\begin{align*}B_{C,\delta}^{x,1}= B_{C,\delta}^1 &= \bigcup_{i \in C} \Big\{\boldsymbol{x}\in\mathbb{R}^d_+\,:\, x_i = 1, x_j \geq 1 \forall j \in C \setminus i, x_k \leq \delta \min_{l \in C} x_l = \delta \forall k \in D \setminus C \Big\}, \\B_{C,\delta}^{x,\delta} &= \bigcup_{i \in D \setminus C} \Big\{\boldsymbol{x}\in\mathbb{R}^d_+\,:\, x_i = \delta\min_{l \in C} x_l, x_j \leq \delta\min_{l \in C} x_l \forall j \in (D\setminus C) \setminus i, x_k \geq 1 \forall k \in C \Big\},\end{align*}

and we specifically note the equality $ B_{C,\delta}^{x,1} = B_{C,\delta}^{1}$ .

Proposition 3.8. Assume the conditions of Proposition 3.5. If

\begin{equation*}{\mathbb P} \left(\!\min_{i \in C} X_{P,i}> t , \max_{j \in D \setminus C} X_{P,j}< t^{\delta} \right) \in \mathrm{RV}^{\infty}_{-1/\tau_{C}(\delta)}\end{equation*}

and

\begin{equation*}{\mathbb P} \left(\!\min_{i \in C} X_{P,i}> t , \max_{j \in D \setminus C} X_{P,j}< \left(\!\min_{i \in C} X_{P,i} \right)^{\delta} \right) \in \mathrm{RV}_{-1/\tilde{\tau}_C(\delta)}^\infty,\end{equation*}

then $ \tilde{\tau}_C(\delta) = \tau_C(\delta)$ .

Proof. Define

\begin{equation*}r_{C,\delta}^x = \min\left\{r \in [0,1] \,:\, r R_{C,\delta}^x \cap G =\emptyset \right\},\end{equation*}

where for $ r>0$ , $ rR_{C,\delta}^x = \big\{\boldsymbol{x}\in \mathbb{R}^d_+\,:\, x_i>r, i \in C,x_j \leq \delta \min_{l \in C} x_l, j \in D\setminus C\big\}$ . Similarly to Propositions 3.3 and 3.5, we have

\begin{align*}{-}\log {\mathbb P}\Big({\boldsymbol{X}}_E \in tr_{C,\delta}^x R_{C,\delta}^x\Big) = {-}\log {\mathbb P}\bigg(\!\min_{i \in C} X_{E,i}>tr_{C,\delta}^x, \max_{j \in D \setminus C} X_{E,j}< \delta \min_{i \in C} X_{E,i}\bigg) \sim t,\end{align*}

and we conclude $ \tilde{\tau}_C(\delta) = r_{C,\delta}^{x}$ . As in Proposition 3.7, we have $ r_{C,\delta}^{x}=\min_{\boldsymbol{y} \in B_{C,\delta}^x} g(\boldsymbol{y})$ . Noting again that $ \arg\min_{\boldsymbol{y} \in B_{C,\delta}^x} g(\boldsymbol{y}) \in B_{C,\delta}^{x,1}=B_{C,\delta}^1$ shows that $ r_{C,\delta}^{x} = r_{C,\delta} = \tau_{C}(\delta)$ .

4.1.1. Meta-Gaussian distribution: nonnegative correlation

Starting with a Gaussian bivariate random vector and transforming its margins to standard exponential, we obtain a meta-Gaussian distribution with exponential margins. Such a distribution inherits the copula of the Gaussian distribution. For simplicity, we consider the case where the underlying Gaussian random vector has standard normal components with correlation $ \rho$ .

Then, for $ \rho\ge0$ , the joint probability density $ f_E$ satisfies

\begin{align*}{-}\log f_E(tx,ty)/t = \big(x + y -2\rho(xy)^{1/2}\big)/\big(1-\rho^2\big) + O(\!\log t/t),\quad t\to\infty,\quad x,y\ge 0,\end{align*}

so that $ g(x,y) = \big(x + y -2\rho(xy)^{1/2}\big)/\big(1-\rho^2\big)$ . The convergence in (2.10) holds on $ [0,\infty)^d$ , and hence the limit set exists and is given by $ \{\boldsymbol{x}\in[0,\infty)^d\,:\, g(\boldsymbol{x})\le1\}$ . This is Case (ii) in Figure 2.

Conditional extremes: Setting $ g(\alpha,1) = 1$ leads to $ \big(\alpha^{1/2} - \rho\big)^2 = 0$ , i.e., $ \alpha = \rho^2$ . For $ \beta$ we have $ g\big(\rho^2+u, 1\big) - 1 = u^2/\big\{2\rho\big(1-\rho^2\big)\big\} + O\big(u^3\big) \in \mathrm{RV}_2^0$ ; hence $ \beta = 1/2$ .

Function $ \lambda(\boldsymbol{\omega})$ : By Proposition 3.4, we need to find $ 1/r_{\omega} = \min_{\boldsymbol{y} \in B_{\omega}} g(x,y)$ . If $ \min\!(\omega,$ $ 1-\omega)/\max\!(\omega,1-\omega) \leq \rho^2$ , then $ \min_{\boldsymbol{y} \in B_{\omega}} g(x,y) = 1$ , with the minima occuring at the points $ (1,\rho^2)$ , $ (\rho^2,1)$ . Otherwise, if $ \min\!(\omega,1-\omega)/\max\!(\omega,1-\omega) \geq \rho^2$ , then $ \min_{\boldsymbol{y} \in B_{\omega}} g(x,y) = g(1, \min\!(\omega,1-\omega)/\max\!(\omega,1-\omega))$ . Putting this together with Proposition 3.3, we find

\begin{equation*}\lambda(\omega,1-\omega) = \begin{cases}\max\!(\omega,1-\omega), & \min\!(\omega,1-\omega)/\max\!(\omega,1-\omega) \leq \rho^2,\\g(\omega,1-\omega) = \frac{1 -2\rho(\omega(1-\omega))^{1/2}}{1-\rho^2}, & \min\!(\omega,1-\omega)/\max\!(\omega,1-\omega) \geq \rho^2.\end{cases}\end{equation*}

This is the same form as given in [Reference Wadsworth and Tawn44]. We therefore have $ \eta_{1,2} = [2g(1/2,1/2)]^{-1} = g(1, 1)^{-1} = (1+\rho)/2$ .

Coefficients $ \tau_j(\delta)$ : From Proposition 3.6, we have $ \tau_1(\delta) = \big[\!\min_{\gamma \in [0,\delta]} g(1,\gamma)\big]^{-1} = \big[g\big(1,\min\big(\delta,\rho^2\big)\big)\big]^{-1}$ . Therefore, $ \tau_{1}(\delta) = 1$ if $ \delta \geq \rho^2$ , and otherwise $ \tau_{1}(\delta) = (1-\rho^2)/\big(1+\delta-2\rho\delta^{1/2}\big) < 1$ . Note that these values are very laborious to calculate via Gaussian survival functions, and they were not given in [Reference Simpson, Wadsworth and Tawn42].

4.1.2. Meta-Gaussian distribution: negative correlation

When $ \rho<0$ , Proposition 2.2 cannot be applied as the continuous convergence condition (2.10) does not hold along the axes. Hence we gain only a partial specification, when $ x>0$ , $ y>0$ , through this route. Instead, here we can apply Proposition 2.3, since the limit function g in (2.13) satisfies the monotonicity condition given immediately thereafter. This limit function is given by

\begin{align*}g(x,y) = \begin{cases}\big(x + y -2\rho(xy)^{1/2}\big)/\big(1-\rho^2\big), & x>0, y>0,\\x, & y=0,\\y, & x=0.\end{cases}\end{align*}

Figure 5 illustrates the limit sets G for the three cases $ \rho>0$ , $ \rho=0$ , and $ \rho<0$ . In the latter case, large values of one variable tend to occur with small values of the other, which causes the limit set to include lines along the axes, and the function g is not continuous. Such difficulties can be alleviated by consideration of Laplace margins for distributions displaying negative dependence, which is discussed further in Section 5.

Figure 5.

Top row: limit sets $ \{(x,y)\,:\, g(x,y)\le1\}$ for a bivariate meta-Gaussian distribution with exponential margins. Bottom row: corresponding plots of function g(x,y) for a fixed value of y.

4.1.3. Logistic generalized Pareto copula

The logistic generalized Pareto distribution with conditionally exponential margins $ \big({\mathbb P}\big(X_{\tilde{E}}>x\big) = {\mathbb P}\big(X_{\tilde{E}}>0\big)e^{-x}, x>0\big)$ and dependence parameter $ \theta \in(0,1)$ satisfies

\begin{align*}f_{\tilde{E}}(x,y) &= \theta^{-1}2^{-\theta} e^{-(x+y)/\theta}\big(e^{-x/\theta}+e^{-y/\theta}\big)^{\theta-2},\\{-}\log f_{\tilde{E}}(tx,ty)/t &= \theta^{-1}\max\!(x,y) + \big(1-\theta^{-1}\big)\min\!(x,y) + O(1/t),\end{align*}

so the gauge function is $ g(x,y) = \theta^{-1}\max\!(x,y) + \big(1-\theta^{-1}\big)\min\!(x,y)$ . This form of gauge function is found throughout several symmetric asymptotically dependent examples, such as those distributions whose spectral measure H places no mass on 0 and 1 and whose densities are regularly varying at the endpoints 0, 1, so that $ \mathrm{d}H(w)/\mathrm{d}w \in \mathrm{RV}_{1/\theta-2}^{0}$ , $ -\mathrm{d}H(1-w)/\mathrm{d}w \in \mathrm{RV}_{1/\theta-2}^{0}$ . This is Case (i) in Figure 2.

Conditional extremes: Solving for $ g(\alpha,1)=1$ , we obtain $ \alpha=1$ , whilst $ g(1+u,1) -1 = u/\theta \in \mathrm{RV}_1^0$ , and hence $ \beta=0$ .

Function $ \lambda(\boldsymbol{\omega})$ : We have that $ \arg\min_{\boldsymbol{y} \in B_\omega}g(\boldsymbol{y}) = (1, 1)$ , so $ r_{ \omega} = 1$ and $ \lambda(\omega,1-\omega) = \max\!(\omega,1-\omega)$ . Therefore $ \eta_{1,2} =1$ .

Coefficients $ \tau_j(\delta)$ : We have $ \tau_1(\delta) = [\min_{\gamma \in [0,\delta]} g(1,\gamma)]^{-1} = [g(1,\delta)]^{-1} = [\theta^{-1}+(1-\theta^{-1}\delta)]^{-1}$ . This matches the value calculated in the Supplementary Material of [Reference Simpson, Wadsworth and Tawn42].

4.1.4. Inverted extreme value distribution

The inverted extreme value copula is the joint lower tail of an extreme value copula, translated to be the joint upper tail. That is, if $ (U_1,U_2)$ have an extreme value copula with uniform margins, then $ (1-U_1,1-U_2)$ have an inverted extreme value copula. In two dimensions, its density in exponential margins may be expressed as

\begin{equation*}f_E(x,y) = \{l_{1}(x,y)l_2(x,y)-l_{12}(x,y)\}\exp\{-l(x,y)\},\end{equation*}

where $ l(\boldsymbol{x}) = V(1/\boldsymbol{x})$ , for V the exponent function in (2.3), is the 1-homogeneous stable tail dependence function (e.g. [9, Chapter 8]) of the corresponding extreme value distribution, and $ l_1(x,y) = \partial l(x,y) /\partial x$ , etc. We thus have

\begin{equation*}{-}\log f_E(tx,ty)/t = l(x,y) + O(1/t),\qquad t \to\infty,\end{equation*}

so $ g(x,y) = l(x,y)$ .

Conditional extremes: Stable tail dependence functions always satisfy $ l(x,0) = x$ , $ l(0,y) = y$ , and so $ g(1,0) = g(0,1) = 1$ . Hence, if $ \alpha=0$ is the only solution to $ g(\alpha,1) =1$ , then $ a(x)/x \sim 0$ . An example of this is given by the inverted extreme value logistic copula, whereby $ l(x,y) = \big(x^{1/\theta}+y^{1/\theta}\big)^\theta, \theta \in (0,1]$ . This is Case (iii) of Figure 2, for which we have $ \alpha=0$ and $ \beta=1-\theta$ .

Further examples in this class are given in the arXiv version of this paper.

Function $ \lambda(\boldsymbol{\omega})$ : Since $ g(x,y) = l(x,y)$ , and l is a convex function satisfying $ l(x,0)=x$ , $ l(0,y) = y$ , we have $ \arg\min_{\boldsymbol{y} \in B_{\omega}} g(\boldsymbol{y}) = (\omega,1-\omega)/\max\!(\omega,1-\omega)$ . Hence, $ \lambda(\omega,1-\omega) = g(\omega,1-\omega)$ in this case.

Coefficients $ \tau_j(\delta)$ : Since $ g(1,0)=1$ , we have $ \tau_1(\delta) = 1$ for all $ \delta \in [0,1]$ .

4.1.5. Density defined by g

If $ g\,:\,\mathbb{R}^d_+ \to \mathbb{R}_+$ is a gauge function describing a limit set G, then $ f(\boldsymbol{x}) = e^{-g(\boldsymbol{x})}/(d! |G|)$ is a density (see [Reference Balkema and Nolde4]). In general, except for the case of $ g(\boldsymbol{x}) = \sum_{i=1}^d x_i$ , the margins are not exactly exponential, and may be heavier than exponential, for example in the case $ g(\boldsymbol{x}) = \max_{1\leq i \leq d}(x_i)$ .

We consider the density defined by $ g(x,y) = \max\{(x-y)/\theta,(y-x)/\theta,(x+y)/(2-\theta)\}$ , $ \theta \in (0,1]$ : this is Case (vi) in Figure 2, and illustrated in Figure 4. The marginal density is given by

\begin{equation*}\Big[2 e^{-x} - \theta e^{-x/\theta} -2(1-\theta)e^{-x/(1-\theta)}\Big]/\big[4\theta - 3\theta^2\big].\end{equation*}

Conditional extremes: Solving for $ g(\alpha,1)=1$ , we obtain $ \alpha=1-\theta$ , whilst $ g(1-\theta+u,1) -1 = u/(2-\theta) \in \mathrm{RV}_1^0$ , and hence $ \beta=0$ .

Function $ \lambda(\boldsymbol{\omega})$ : If $ \min\!(\omega,1-\omega)/\max\!(\omega,1-\omega) \leq 1-\theta$ , then $ \arg\min_{\boldsymbol{y} \in B_\omega} = (1,1-\theta)$ , or $ (1-\theta,1)$ and $ r_{\omega}=1$ ; otherwise, $ \arg\min_{\boldsymbol{y} \in B_\omega} = (1,\omega/(1-\omega))$ or $ ((1-\omega)/\omega, 1)$ , and $ r_{\omega} =\{1+ \min\!(\omega,1-\omega)/\max\!(\omega,1-\omega)/(2-\theta)\}$ . Therefore,

\begin{equation*}\lambda(\omega,1-\omega) = \begin{cases}\max\!(\omega,1-\omega), & \min\!(\omega,1-\omega)/\max\!(\omega,1-\omega) \leq 1-\theta,\\g(\omega,1-\omega) = \frac{1}{2-\theta}, & \min\!(\omega,1-\omega)/\max\!(\omega,1-\omega) \geq 1-\theta,\end{cases}\end{equation*}

and the residual tail dependence coefficient $ \eta_{1,2} = 1-\theta/2$ .

Coefficients $ \tau_j(\delta)$ : We have $ \tau_1(\delta) = [\!\min_{\gamma \in [0,\delta]} g(1,\gamma)]^{-1} = [g(1,\min\!(\delta,1-\theta))]^{-1}$ . Therefore, $ \tau_{1}(\delta) = 1$ if $ \delta \geq 1-\theta$ ; otherwise, $ \tau_{1}(\delta) = \theta/(1-\delta)< 1$ .

Further examples presented in the arXiv version include the Hüsler–Reiss generalized Pareto copula and two boundary cases with $ g(x,y)=\max\!(x,y)$ , one displaying asymptotic dependence and one asymptotic independence. In the latter of these, we find that there is no conditional extreme value limit with positive support, but there is one with negative support. We comment that the results of Proposition 3.1 do indeed focus predominantly on the positive end of the support for limit distributions, but most known examples of conditional limits have support including $ (0,\infty)$ . A natural next step is to consider the implications relating to negative support. We particularly note the possibility that the orders of regular variation of the two functions $ g(1,\alpha_1+u)-1 \in \mathrm{RV}^0_{1/\big(1-\beta_1^+\big)}$ and $ g\big(1,\alpha_1-u\big)-1\in \mathrm{RV}^0_{1/\big(1-\beta_1^-\big)}$ need not be equal, though for each of our examples where both functions are regularly varying, $ \beta^+=\beta^-$ . If $ \beta^+>\beta^-$ , it seems likely that a limit distribution with positive support only would arise, and vice versa when $ \beta^+<\beta^-$ .

4.2.1. Gaussian copula

The general form of the gauge function for a meta-Gaussian distribution with standard exponential margins and correlation matrix $ \Sigma$ with nonnegative entries is

\begin{equation*}g(\boldsymbol{x}) = \big(\boldsymbol{x}^{1/2}\big)^\top \Sigma^{-1} \boldsymbol{x}^{1/2}.\end{equation*}

Figure 6 displays the level set $ g(\boldsymbol{x}) = 1$ when the Gaussian correlations in $ \Sigma$ are $ \rho_{12}=0.75$ , $ \rho_{13}=0.25$ , $ \rho_{23} =0.4$ . The red dots on the level set are the points $ (1,1,\gamma)/g(1,1,\gamma)$ , $ (1,\gamma,1)/g(1,\gamma,1)$ , and $ (\gamma,1,1)/g(\gamma,1,1)$ for $ \gamma \in [0,1]$ . The figure also provides an illustration of $ \tau_{2,3}(\delta)$ for $ \delta =0.2$ and $ \delta=0.8$ : in each case the light blue line from the origin is $ \gamma \times (\delta,1,1)$ , $ \gamma \in [0,1]$ , whilst the pink lines trace out the boundary $ B_{\{2,3\},\delta}$ and $ \tau_{2,3}(\delta)B_{\{2,3\},\delta}$ . We see that when $ \delta = 0.2$ (left panel), $ \tau_{2,3}(0.2) = 1/g(0.2,1,1)$ , i.e., $ \min_{\boldsymbol{y} \in B_{\{2,3\},0.2}}g(\boldsymbol{y}) = g(0.2,1,1)$ . However, when $ \delta = 0.8$ , $ \min_{\boldsymbol{y} \in B_{\{2,3\},0.8}}g(\boldsymbol{y}) = g(\gamma^\star,1,1)$ , for $ \gamma^\star \in [0,0.8]$ , so $ \tau_{2,3}(0.8) = 1/g(\gamma^\star,1,1)$ . We note that the same value of $ \tau_{2,3}(\delta)$ applies for any $ \delta \geq \gamma^\star$ : for this example, when $ \delta \geq \gamma^\star \approx 0.51$ , $ \tau_{2,3}(\delta) = 0.7 = \eta_{2,3}$ .

Figure 6.

Level set $ g(\boldsymbol{x})=1$ for a trivariate meta-Gaussian distribution with exponential margins. The left panel illustrates $ \tau_{2,3}(0.2)$ : the boundary set indicated by the pink solid lines is scaled along the blue trajectory emanating from the origin until it touches G, which happens in this case at the corner point $ (0.2,1,1)/g(0.2,1,1)$ . The centre panel illustrates $ \tau_{2,3}(0.8)$ : the boundary set is again pulled back along the indicated trajectory until it touches G; in this case this does not occur at a corner point. The right panel illustrates $ \tau_1(\delta)$ in a similar manner, for $ \delta =0.2,0.6$ .

The reason that $ \tau_{2,3}(\delta) = \eta_{2,3}$ for sufficiently large $ \delta$ is that, in this case, $ \arg\min_{x_1} g(x_1,1,1) = \gamma^\star,$ meaning that the two-dimensional marginalization $ g_{\{2,3\}}(1, 1) = g(\gamma^\star,1,1)$ , and we further have that $ g_{\{2,3\}}(1, 1) = \min_{\boldsymbol{y} \in B_{2,3}} g_{\{2,3\}}(\boldsymbol{y})$ , so $ \eta_{2,3} = 1/g_{2,3}(1, 1)$ . In Section 4.2.2 we will illustrate a gauge function for which $ \arg\min_{x_3} g(1,1,x_3)>1$ , and consequently $ \tau_{1,2}(\delta) < \eta_{1,2}$ for all $ \delta\leq 1$ .

The right panel of Figure 6 illustrates $ \tau_{1}(\delta)$ for $ \delta = 0.2$ and $ \delta =0.6$ . When $ \delta =0.6$ , the boundary $ B_{1,\delta}$ already touches G, and so $ \tau_{1}(0.6)=1$ . In this example, $ \tau_1(\delta) =1$ for any $ \delta \geq 0.5625 = \rho_{12}^2$ . Thus, $ \tau_{1}(0.2)<1$ as illustrated in the figure. We comment that if we had marginalized over $ X_2$ , and were looking at $ \tau_1(\delta)$ for the variables $ (X_1,X_3)$ , then we would have $ \tau_1(\delta) = 1$ for any $ \delta \geq 0.0625 = \rho_{13}^2$ . This provides an illustration of the dimensionality of the problem interacting with $ \tau_{C}(\delta)$ , and is again related to the point at which the minimum point defining the lower-dimensional gauge function occurs.

4.2.2. Vine copula

Three-dimensional vine copulas are specified by three bivariate copulas: two in the ‘base layer’, giving the dependence between, e.g., $ X_1,X_2$ and $ X_2,X_3$ , and a further copula specifying the dependence between $ X_1|X_2$ and $ X_3|X_2$ . Here we take the base copulas to be independence for $ (X_1,X_2)$ , and the inverted Clayton copula with parameter $ \beta>0$ for $ (X_2,X_3)$ . The final copula is taken as inverted Clayton with parameter $ \gamma>0$ . The gauge function that arises in exponential margins is

(4.1) \begin{align}g(\boldsymbol{x}) = (1+\beta)\max\!(x_2,x_3) &-\beta \min\!(x_2,x_3) -\gamma x_1-(\gamma+1)(\beta+1)(\max\!(x_2,x_3)-x_2) \notag \\&+ (2\gamma+1)\max\!(x_1,(\beta+1)(\max\!(x_2,x_3)-x_2)).\end{align}

Figure 7 displays the level set $ g(\boldsymbol{x})=1$ . In this figure we also give an illustration of a case where $ \tau_C(1)<\eta_C$ : in particular, for this example $ \tau_{1,2}(1)<\eta_{1,2,3}=\eta_{1,2}=1/2$ . The purple lines (near right of image) represent the boundary of the region $ \tau_{1,2}(1)R_{\{1,2\},1} = \tau_{1,2}(1,\infty]^2 \times [0,1]$ , while the green lines (back right of image) represent the boundary of the region $ \eta_{1,2,3}(1,\infty]^3$ . Theorem 1 of [Reference Simpson, Wadsworth and Tawn42] tells us that $ \eta_{1,2} = \max\!(\tau_{1,2}(1),\tau_{1,2,3})$ , where $ \tau_{1,2,3}=\eta_{1,2,3}$ . Therefore $ \tau_{1,2}(1)<\eta_{1,2}$ guarantees that $ \eta_{1,2}=\eta_{1,2,3}$ .

Figure 7.

Level set $ g(\boldsymbol{x})=1$ with g as in (4.1). The figure illustrates $ \tau_{1,2}(1)$ and $ \eta_{1,2,3} = \eta_{1,2} = 1/2$ .

We also illustrate Proposition 2.4, minimizing (4.1) over $ x_3$ . If $ x_2>x_3$ then the minimum over $ x_3$ occurs if we set $ x_3=x_2$ and is equal to $ x_2+(1+\gamma)x_1$ . If $ x_2< x_3$ then owing to the final term we need to consider the cases $ x_3 \lessgtr x_1/(1+\beta)+x_2$ . In both cases, the minimum is attained at $ x_3 = x_1/(1+\beta)+x_2$ , and is equal to $ x_1+x_2 < (1+\gamma)x_1+x_2$ . Therefore, $ \min_{x_3} g(x_1,x_2,x_3) = x_1+x_2$ . This result is as expected since the bivariate margins of vine copulas that are directly specified in the base layer are equal to the specified copula: in this case, independence.