On the structure of exchangeable extreme-value copulas

Abstract

We show that the set of $d$-variate symmetric stable tail dependence functions is a simplex and we determine its extremal boundary. The subset of elements which arises as $d$-margins of the set of $(d+k)$-variate symmetric stable tail dependence functions is shown to be proper for arbitrary $k\ge 1$. Finally, we derive an intuitive and useful necessary condition for a bivariate extreme-value copula to arise as bi-margin of an exchangeable extreme-value copula of arbitrarily large dimension, and thus to be conditionally iid.

Introduction

The problem of determining whether a given exchangeable probability law on ${\mathbb{R}}^d$ can arise as a $d$-dimensional margin of some exchangeable probability law on ${\mathbb{R}}^{d+k}$, $k\ge 1$, is known as the extendibility problem in the literature. If there is a solution to the extendibility problem for arbitrary $k\ge 1$, one calls the probability law (infinitely) extendible. In the general case, that is without postulating any additional conditions on the involved probability distributions, [12] derives an analytical criterion to check for extendibility, although this criterion is difficult to apply in concrete cases. Analytical solutions of the infinite extendibility problem have natural connections with harmonic analysis, rendering the topic interesting for theorists. But the problem is also interesting for applied probabilists, since infinitely extendible models can be used as flexible dependence models that are still very convenient to work with in large dimensions. The most famous solutions to the infinite extendibility problem for specific families of distributions comprise ${\ell }_2$-norm symmetric laws (Schoenberg’s Theorem), ${\ell }_1$-norm symmetric laws associated with Archimedean copulas (see [18]), and ${\ell }_{\infty }$-norm symmetric laws (see [8]), see also [21] for a nice wrapping of these three popular cases and a generalization to ${\ell }_p$-symmetric laws for arbitrary $p\in (0,\infty ]$. More recently, the infinite extendibility problem has also been solved for popular families of multivariate exponential and geometric distributions, see [15], [17], and has also been dealt with for extreme-value distributions in [13], [16]. Recall further that the seminal de Finetti Theorem implies that the notions “infinitely extendible” and “conditionally iid” coincide for exchangeable probability laws, see [3].

The present article may be seen as a continuation of the work in [13], [16] and deals with the extendibility problem for exchangeable extreme-value copulas. More specifically, we investigate which exchangeable $d$-variate extreme-value copulas arise as $d$-margins of some $(d+k)$-variate exchangeable extreme-value copula, and which do not. Recall that an extreme-value copula $C:{[0,1]}^d\to [0,1]$ is (the restriction to ${[0,1]}^d$ of) a distribution function with one-dimensional margins that are uniform on $[0,1]$, and which satisfies the extreme-value property $C(u_1^t,\dots ,u_d^t)=C{(u_1,\dots ,u_d)}^t$ for all $u_1,\dots ,u_d\in [0,1]$ and $t\ge 0$. The extreme-value property analytically characterizes multivariate distribution functions that can arise as limits of appropriately normalized componentwise maxima/minima of independent and identically distributed random vectors, see [22] for a textbook account on multivariate extreme-value theory. The restriction to uniform one-dimensional margins, i.e. to copulas, instead of arbitrary extreme-value distribution functions is without loss of generality, since by virtue of Sklar’s Theorem we can write an arbitrary distribution function of a $d$-variate extreme-value distribution as $F(x_1,\dots ,x_d)≔C\left\{F_1\left(x_1\right),\dots ,F\left(x_d\right)\right\}$, where $C:{[0,1]}^d\to [0,1]$ is an extreme-value copula and $F_1,\dots ,F_d$ denote the one-dimensional margins, which are necessarily one-dimensional extreme-value distribution functions. We refer to [6], [11] for general background on copulas, and to [10] for a book on copulas with a specific chapter on extreme-value copulas.

Each $d$-dimensional extreme-value copula is uniquely associated with its so-called ‘stable tail dependence function’ $\ell :{[0,\infty )}^d\to [0,\infty )$, given by

$$\ell \left(\overrightarrow{x}\right)=-ln\left\{C\left(e^{-x_1},\dots ,e^{-x_d}\right)\right\},$$

where arrows indicate vectors, i.e. $\overrightarrow{x}=(x_1,\dots ,x_d)$. The extreme-value property of $C$ implies that $\ell$ is homogeneous of order one, i.e. $\ell \left(t\overrightarrow{x}\right)=t\ell \left(\overrightarrow{x}\right)$. In fact, each stable tail dependence function $\ell$ even equals the restriction of an orthant-monotonic norm on ${\mathbb{R}}^d$ to ${[0,\infty )}^d$, but not every orthant-monotonic norm defines a stable tail dependence function for $d\ge 3$ (only for $d=2$), see [19] for background on the matter and [7] for a textbook account. While a norm is convex, stable tail dependence functions satisfy the in general stronger notion of (fully) $d$-max-decreasingness, see [23]. Due to the homogeneity property, the restriction of $\ell$ to the $d$-dimensional unit simplex $S_d≔\{\overrightarrow{x}\in {[0,1]}^d:x_1+\cdots +x_d=1\}$ already determines $C$. Further, there exists a random vector $\overrightarrow{Q}=(Q_1,\dots ,Q_d)$, taking values in $S_d$ and satisfying $\mathrm{E}\left[Q_k\right]=1∕d$ for each component $k$, such that

$$\ell \left(\overrightarrow{x}\right)=d\mathrm{E}\left[max\{x_1{Q}_1,\dots ,x_d{Q}_d\}\right].$$

This random vector $\overrightarrow{Q}$ is uniquely determined in distribution, see [4], [23], and the finite measure $dPr(\overrightarrow{Q}\in \mathrm{d}q)$ on $S_d$ is called ‘Pickands dependence measure’, named after [20]. Conversely, any random vector $\overrightarrow{Q}=(Q_1,\dots ,Q_d)$ with values in $S_d$ and the property $\mathrm{E}\left(Q_k\right)=1∕d$ for each component $k$ defines a valid stable tail dependence function via Eq. (1), see [9]. The stable tail dependence function $\ell$ is symmetric in its arguments if and only if the random vector $\overrightarrow{Q}$ is exchangeable, see the proof of Lemma 2. In the light of these notations, the present article deals with the analytical question whether a symmetric stable tail dependence function $\ell$ arises as the $d$-variate margin of some higher-dimensional object of the same structural form, i.e. whether its dimension can be extended or not.

Regarding the important special case of infinite extendibility, we say that a $d$-variate exchangeable extreme-value copula $C$ is infinitely extendible if it arises as $d$-margin of some $(d+k)$-dimensional exchangeable extreme-value copula for arbitrary $k\ge 1$. [16] shows that this is the case if and only if there is a right-continuous, non-decreasing stochastic process $H={\left\{H_t\right\}}_{t\ge 0}$ satisfying $H_0=0$, ${lim}_{t\to \infty }{H}_t=\infty$, and $-ln\left[\mathrm{E}\{exp(-H_1)\}\right]=1$, which is strongly infinitely divisible with respect to time, such that $C\left(\overrightarrow{u}\right)=Pr(U_1\le u_1,\dots ,U_d\le u_d)$ for $\overrightarrow{u}=(u_1,\dots ,u_d)\in {[0,1]}^d$, where

$$U_k≔exp(-X_k),X_k≔inf\{t>0:H_t>{ϵ}_k\},k\in \{1,\dots ,d\},$$

and ${ϵ}_1,\dots ,{ϵ}_d$ are independent and identically distributed with standard exponential distribution, independent of $H$. ‘Infinite divisibility with respect to time’ means that for arbitrary $n\in \mathbb{N}$ the stochastic process $H$ is identical in law with the stochastic process $H_{.∕n}^{\left(1\right)}+\cdots +H_{.∕n}^{\left(n\right)}$ for independent copies $H^{\left(1\right)},\dots ,H^{\left(n\right)}$ of $H$. Making use of this result, [13] further shows that $C$ is infinitely extendible if and only if there is a pair $(b,\lambda )$ comprising a constant $b\in [0,1]$ and a probability measure $\lambda$ on the set ${\mathfrak{F}}_1$ of distribution functions of non-negative random variables with unit mean such that the associated stable tail dependence function is given by

$$\ell \left(\overrightarrow{x}\right)=b\sum _{k=1}^d{x}_k+(1-b){\int }_{{\mathfrak{F}}_1}{\int }_0^{\infty }1-\prod _{k=1}^{d}F\left(\frac{s}{x_k}\right)\mathrm{d}s\lambda \left(\mathrm{d}F\right).$$

The pair $(b,\lambda )$ can be used to specify the process $H$ in (2) as

$$H_t=bt+(1-b)\sum _{k\ge 1}-ln\left\{G_k\left(\frac{{\eta }_1+\cdots +{\eta }_k}{t}-\right)\right\},t\ge 0,$$

where $G_1,G_2,\dots$ is a sequence of independent and identically distributed random distribution functions drawn from ${\mathfrak{F}}_1$ according to $\lambda$ and ${\eta }_1,{\eta }_2,\dots$ is an independent sequence of independent and identically distributed standard exponential variates. The stochastic model (2) for infinitely extendible extreme-value copulas is very convenient for applications due to the fact that $U_1,\dots ,U_d$ are independent and identically distributed conditioned on the $\sigma$-algebra generated by $H$. Unfortunately, it is in general unclear whether a given exchangeable extreme-value copula is of the form in Eq. (3) and thus admits the convenient stochastic representation (2).

With this background in mind, the present article finally provides the answer to a natural question about which the authors were pondering for almost a decade. It has served as a fruitful source of inspiration, since we have been able to discover plenty of related results in the meantime while chasing our ‘Moby Dick’, which is:

Is a symmetric stable tail dependence function always of the form (3), and thus the associated random vector has a stochastic representation as in (2)?

We are going to answer this question with ‘no’ in this article, for arbitrary dimension $d\ge 2$. Whereas for $d=3$ (and from this easily also for $d\ge 3$) already the remark after Example 1 of [13] shows that not every exchangeable extreme-value copula is of the form (3), for $d=2$ this question has been open until now. Our strategy of proof provides a structural result of independent interest for exchangeable extreme-value copulas in arbitrary dimension $d\ge 2$: the set of symmetric stable tail dependence functions, equipped with the topology induced by pointwise convergence, is a simplex. We determine its extremal boundary and show that extremal elements cannot arise as margins of a higher-dimensional symmetric stable tail dependence function.

For the sake of clarity, we formally introduce the notion of a simplex, that we use throughout, referring to [1] for a textbook account on the topic. Recall that the set of real-valued functions defined on some subset of ${\mathbb{R}}^d$ is a locally convex Hausdorff vector space with respect to the topology of pointwise convergence. We view the set of $d$-dimensional stable tail dependence functions ${\mathfrak{L}}_d$ as a subset of this space and observe that this set is convex and compact, likewise the subset ${\mathfrak{L}}_d^X$ of symmetric stable tail dependence functions. Now we denote by $\mathcal{C}$ one of the compact convex sets ${\mathfrak{L}}_d$ or ${\mathfrak{L}}_d^X$. An element $\ell \in \mathcal{C}$ is called ‘extremal’, or lies in the ‘extremal boundary’ ${\partial }_e\mathcal{C}$, if $\ell =\alpha {\ell }_1+(1-\alpha ){\ell }_2$ with $\alpha \in (0,1)$ and ${\ell }_1,{\ell }_2\in \mathcal{C}$ necessarily implies $\ell ={\ell }_1={\ell }_2$. By the Krein–Milman Theorem, each element $\ell \in \mathcal{C}$ has a representation of the form $\ell ={\int }_{{\partial }_e\mathcal{C}}\tilde{\ell }{\nu }_{\ell }\left(\mathrm{d}\tilde{\ell }\right)$ for a probability measure ${\nu }_{\ell }$ defined on the extremal boundary ${\partial }_e\mathcal{C}$. The set $\mathcal{C}$ is called a ‘simplex’ if ${\nu }_{\ell }$ is unique, i.e. if there is a one-to-one relation between $\ell$ and ${\nu }_{\ell }$. The boundary ${\partial }_e{\mathfrak{L}}_2$ is computed in [14], [26]. [2] points out, by providing a non-exchangeable example, that ${\mathfrak{L}}_2$ is not a simplex. The present article shows that ${\mathfrak{L}}_d^X$ is indeed a simplex and determines the extremal boundary ${\partial }_e{\mathfrak{L}}_d^X$.

It is well known that infinitely extendible random vectors are necessarily positively associated in some sense. In fact, [24] even calls such random vectors ‘positive dependent by mixture’ (PDM), which is a third synonym that is found in the literature for “conditionally iid” or “infinitely extendible”. Thus, a popular strategy to prove that a random vector is not infinitely extendible is to show that its components exhibit some sort of negative association, which is successful for many popular families of distributions for which exchangeable but not infinitely extendible examples are known. What makes the investigation of extreme-value copulas more delicate in this regard is the fact that extreme-value copulas always induce non-negative association, see [10, Theorem 6.7, p. 177], which disqualifies the aforementioned strategy of proof. Furthermore, well-known parametric families of exchangeable bivariate extreme-value copulas, like the Gumbel copula, the Galambos copula, the Hüsler–Reiss copula, or the Cuadras–Augé copula, are indeed of the form (3). Note that the Cuadras–Augé copula equals the exchangeable, bivariate Marshall–Olkin copula, see also Example 2.

The property of being infinitely extendible has far-reaching consequences in applications. If a model is not infinitely extendible, it is difficult, or even impossible, to add more components to the existing model while preserving its structure. There are many practical applications where it is natural and required to let the dimension $d$ vary. Consider, for example, an insurance company (resp. a bank) that has $d$ insured objects under contract (resp. has issued $d$ loans). In such applications, the quantity $d$ changes frequently in a natural way and it would be inconvenient or impracticable to use a model that is limited to an upper bound for $d$. Moreover, a conditionally iid structure allows to apply classical limit theorems like the law of large numbers or Glivenko–Cantelli in a conditional version, which can be a valuable tool in applications.

From a theoretical perspective the extendibility problem is interesting and challenging, as the canonical stochastic model of the random vector in concern might not at all be related to the concept of conditional independence. The stochastic model (2) is by definition based on a two-step Bayesian simulation: first simulate the latent factor $H$, second simulate an iid sequence drawn from the distribution function $1-exp(-H)$. In contrast, the canonical stochastic model for arbitrary (not necessarily exchangeable or even infinitely extendible) extreme-value copulas, based on limits of component-wise minima/maxima, does a priori not have any connection to conditional independence.

Section 2 is concerned with the algebraic structure of exchangeable $d$-variate extreme-value copulas. The main contribution of Sections 2.1 Bivariate extreme-value copulas: Characterization and geometry, 2.2 Multivariate extreme-value copulas: Characterization and geometry is the observation that the associated family of symmetric stable tail dependence functions forms a simplex. In Section 2.3 it is shown that certain extremal stable tail dependence functions cannot arise as lower-dimensional margins of higher-dimensional symmetric stable tail dependence functions, thus proving that the notions “infinitely extendible” and “exchangeable” are not synonyms in the realm of extreme-value copulas. Section 3 provides a non-trivial and intuitive necessary condition for a bivariate extreme-value copula to be infinitely extendible.

Throughout, we denote equality in distribution by $\stackrel{d}{=}$, the symbol $\sim$ means “is distributed according to”, and the acronym iid means independent and identically distributed. We say that a probability measure $\mu$ on ${\mathbb{R}}^d$ is ‘conditionally iid’ if there is a random vector $\overrightarrow{X}=(X_1,\dots ,X_d)\sim \mu$ on some probability space $(\Omega ,\mathcal{F},Pr)$ such that $X_1,\dots ,X_d$ are iid conditioned on some $\sigma$-algebra $\mathcal{T}\subset \mathcal{F}$. In this case, we also say that $\overrightarrow{X}$ itself and also its distribution function are conditionally iid. Conditioned on $\mathcal{T}$, and thus also unconditioned, the limiting process $d\to \infty$ is clearly viable by the natural product space extension, and thus we may view $(X_1,\dots ,X_d)$ as the first $d$ members of an infinite sequence ${\left\{X_k\right\}}_{k\in \mathbb{N}}$ whose components are iid conditioned on $\mathcal{T}$. Precisely for this reason, the notion “conditionally iid” is sometimes also referred to as “(infinitely) extendible” in the literature on exchangeable probability laws, for instance in [25].