Exceedance Counts and GOD’s Order Statistics

Abstract

In this paper we derive a characterization of the distribution of the number of exceedances among the components of a random vector in terms of order statistics of generators of D-norms (GOD). The computation of the fragility index is an immediate consequence.

1 Introduction

Let $\boldsymbol {X}=(X_{1},\dots ,X_{d})$ be a random vector that realizes in $\mathbb {R}^{d}$. We are interested in the number of exceedances among the components $X_{1},\dots ,X_{d}$ above high thresholds. Precisely, choose $\boldsymbol {x} = (x_{1},\dots ,x_{d})\in \mathbb {R}^{d}$, $k\in \left \{{1,\dots ,d}\right \}$ and put $N_{\boldsymbol {x}}:={\sum }_{i=1}^{d} 1_{(x_{i},\infty )}(X_{i})$. We want to analyze in this paper the probability

$$ \begin{array}{@{}rcl@{}} P(X_{i}>x_{i}\text{for at least} k \text{of the components} i=1,\dots,d) = P\left( N_{\boldsymbol{x}}\ge k\right) \end{array} $$

for a large x, i.e., each component x_i of x is large.

Suppose the vector $\boldsymbol {x}=(x,\dots ,x)\in \mathbb {R}^{d}$ has constant entry $x\in \mathbb {R}$ and put $N_{x}:=N_{(x,\dots ,x)}$. The fragility index (FI) corresponding to X is the asymptotic conditional expected number of exceedances, given that there is at least one exceedance, i.e., $\text {FI}=\lim _{x\nearrow }E(N_{x}\mid N_{x}>0)$.

The FI was introduced by Geluk et al. (2007) to measure the stability of the stochastic system $\left \{X_{1},\dots ,X_{d}\right \}$. The system is called stable if FI = 1, otherwise it is called fragile.

In Falk and Tichy (2012) the asymptotic conditional distribution $p_{k}:= \lim _{x \nearrow }P(N_{x}=k\mid N_{x}>0)$ was investigated under the condition that the components $X_{1},\dots ,X_{d}$ are identically distributed.

It turned out that this asymptotic conditional distribution of exceedance counts (ACDEC) exists, if the copula C, which corresponds to X by Sklar’s theorem (c.f. Sklar (1959), Sklar (1996)), is in the max-domain of attraction of a multivariate extreme value distribution (EVD) G. This means that, for any $\boldsymbol {x}=(x_{1},\dots ,x_{d})\le \boldsymbol {0}\in \mathbb {R}^{d}$,

$$ C^{n}\left( 1+\frac{x_{1}}n,\dots,1+\frac{x_{d}}n\right)\to_{n\to\infty}G(\boldsymbol{x}), $$

(1.1)

where G is a non degenerate distribution function (df) on $\mathbb {R}^{d}$. This is quite a mild condition, satisfied by almost every textbook copula C, c.f. Section 3.3 in Falk (2019).

Falk and Tichy (2011) investigated the ACDEC, dropping the assumption that the margins $X_{1},\dots ,X_{d}$ are identically distributed.

Recent results on D-norms (Falk (2019), Falk and Fuller (2021)) enable in the present paper the representation of the distribution of N_x in terms of order statistics corresponding to the generator of a D-norm, introduced below.

Let $Z_{1},\dots ,Z_{d}$ be random variables with the properties

$$ Z_{i}\ge 0, E(Z_{i})=1,\qquad 1\le i\le d. $$

Then

$$ \left\Vert\boldsymbol{x}\right\Vert_{D} := E\left( \underset{1\le i\le d}{\max} (\left\vert x_{i}\right\vert Z_{i})\right),\qquad \boldsymbol{x}=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}, $$

defines a norm on $\mathbb {R}^{d}$, called D-norm, and $\boldsymbol {Z}=(Z_{1},\dots ,Z_{d})$ is a generator of the D-norm (GOD).

D-norms are the skeleton of multivariate extreme value theory (MEVT) as elaborated in detail in Falk (2019). Suppose, for example, that the random vector $\boldsymbol {X}=(X_{1},\dots ,X_{d})$ follows a standard multivariate Generalized Pareto distribution (GPD). In this case there exists a D-norm $\left \Vert \cdot \right \Vert _{D}$ on $\mathbb {R}^{d}$ such that

$$ P(\boldsymbol{X}\le\boldsymbol{x}) = 1-\left\Vert\boldsymbol{x}\right\Vert_{D}, $$

for all x in a left neighborhood of $\boldsymbol {0}=(0,\dots ,0)\in \mathbb {R}^{d}$, i.e., for all x ∈ [ε,0]^d with some ε < 0. The characteristic property of a GPD is its excursion stability or exceedance stability, see Proposition 3.1.2 and Remark 3.1.3 in Falk (2019). This stability explains the crucial role of GPD in MEVT. In what follows, all operations on vectors such as x ≤y etc. are always meant componentwise.

According to equation (2.16) in Falk (2019), the survival function of a standard GPD is given by

$$ P(\boldsymbol{X}>\boldsymbol{x})=P(\boldsymbol{X}\ge\boldsymbol{x}) = \wr\!\!\wr\boldsymbol{x}\wr\!\!\wr_{D} := E\left( \underset{1\le i\le d}{\min} (\left\vert x_{i}\right\vert Z_{i})\right),\qquad \boldsymbol{x}\in[\varepsilon,0]^{d}, $$

where ≀ ≀x ≀ ≀_D is the dual D-norm function pertaining to $\left \Vert \cdot \right \Vert _{D}$. Note that the df of X is continuous on [ε,0]^d and, thus, P(X ≥x) = P(X > x) for x ∈ [ε,0]^d.

The generator Z of $\left \Vert \cdot \right \Vert _{D}$ is not uniquely determined. But we know from Corollary 1.6.3 in Falk (2019) that the dual D-norm function ≀ ≀x ≀ ≀_D, which corresponds to $\left \Vert \cdot \right \Vert _{D}$, is independent of the particular generator Z of $\left \Vert \cdot \right \Vert _{D}$.

So far we have considered the maximum $\max \limits _{1\le i\le d}(\left \vert x_{i}\right \vert Z_{i})$ and the minimum $\min \limits _{1\le i\le d}(\left \vert x_{i}\right \vert Z_{i})$, leading to the D-norm $\left \Vert \boldsymbol {x}\right \Vert _{D}$ and dual D-norm ≀ ≀x ≀ ≀_D by taking expectations. But we can clearly order the set $\left \{{\left \vert {x_{1}}\right \vert Z_{1},\dots ,\left \vert {x_{d}}\right \vert Z_{d}}\right \}$ completely

$$ \begin{array}{@{}rcl@{}} \underset{1\le i\le d}{\min} (\left\vert x_{i}\right\vert Z_{i})&\le& \text{2nd smallest value among }\left\{\left\vert x_{1}\right\vert Z_{1},\dots,\left\vert x_{d}\right\vert Z_{d}\right\}\\ &\le& \text{3rd smallest value among }\left\{\left\vert x_{1}\right\vert Z_{1},\dots,\left\vert x_{d}\right\vert Z_{d}\right\}\\ &\vdots&\\ &\le& \underset{1\le i\le d}{\max} (\left\vert x_{i}\right\vert Z_{i}) \end{array} $$

and put, for $k=1,\dots ,d$,

$$ \left\Vert\boldsymbol{x}\right\Vert_{D,(k)}:= E\left( k\text{-th smallest value among }\left\{\left\vert x_{1}\right\vert Z_{1},\dots,\left\vert x_{d}\right\vert Z_{d}\right\}\right). $$

We call this sequence $\left \Vert \boldsymbol {x}\right \Vert _{D,(1)}\le \left \Vert \boldsymbol {x}\right \Vert _{D,(2)}\le {\dots } \le \left \Vert \boldsymbol {x}\right \Vert _{D,(d)}$ of increasing functions ordered D-norms. In particular

$$ \left\Vert\boldsymbol{x}\right\Vert_{D,(1)} =\wr\!\!\wr\boldsymbol{x}\wr\!\!\wr_{D} $$

is the dual D-norm function, and

$$ \left\Vert\boldsymbol{x}\right\Vert_{D,(d)} = \left\Vert\boldsymbol{x}\right\Vert_{D} $$

is the D-norm. Note that only $\left \Vert \boldsymbol {x}\right \Vert _{D,(k)}$ with k = d is actually a norm.

We have, for each $k=1,\dots ,d$,

$$ \left\Vert t\boldsymbol{x}\right\Vert_{D,(k)} = t\left\Vert\boldsymbol{x}\right\Vert_{D,(k)},\qquad t\ge 0, \boldsymbol{x}\in\mathbb{R}^{d}, $$

i.e., $\left \Vert \cdot \right \Vert _{D,(k)}$ is homogeneous of order 1, and it is a continuous function on $\mathbb {R}^{d}$, with $\left \Vert \boldsymbol {0}\right \Vert _{D,(k)}=0$ for each $k=1,\dots ,d$. The following result is, therefore, an immediate consequence of Theorem 3.5 in Falk and Fuller (2021).

Lemma 1.1.

For each $k=1,\dots ,d$, the value $\left \Vert \boldsymbol {x}\right \Vert _{D,(k)}$ does not depend on the choice of the particular generator Z of $\left \Vert \cdot \right \Vert _{D}$.

For a vector with constant entry $\boldsymbol {x}=(x,\dots ,x)\in \mathbb {R}^{d}$ we obtain

$$ \begin{array}{@{}rcl@{}} \left\Vert\boldsymbol{x}\right\Vert_{D,(k)}&=& \left\vert x\right\vert E(k\text{-th smallest value among }\left\{Z_{1},\dots,Z_{d}\right\})\\ &=& \left\vert x\right\vert E\left( Z_{k:d}\right), \end{array} $$

where $Z_{1:d}\le \dots \le Z_{d:d}$ denote the ordered values of $Z_{1},\dots ,Z_{d}$, i.e., the corresponding usual order statistics. This links the analysis of multivariate exceedances with the theory of univariate order statistics.

Though it has received much attention in the literature, the computation of the exact value E(Z_k:d) of an arbitrary order statistic is by no means obvious, even if $Z_{1},\dots ,Z_{d}$ are independent and identically distributed. The exact value is known only in a few cases. Put, for instance, Z_i := 2U_i, 1 ≤ i ≤ d, where $U_{1},\dots ,U_{d}$ are independent random variables with common uniform distribution on (0,1). In this case we have E(Z_k:d) = 2k/(d + 1), see equation (1.7.3) in Reiss (1989).

In the general case, where $Z_{1},\dots ,Z_{d}$ are not necessarily independent and identically distributed, the exact value E(Z_k:d) is in general inaccessible. But a Monte-Carlo simulation would provide an obvious and easy to implement approximation.

The main result in Section 2 is Theorem 2.1, where we establish, for each 1 ≤ k ≤ d, the equation

$$ P(X_{i} > x_{i} \text{for at least} k \text{of the components } 1\le i\le d) = \left\Vert\boldsymbol{x}\right\Vert_{D,(d-k+1)} $$

for x in a left neighborhood of $\boldsymbol {0}\in \mathbb {R}^{d}$, if $\boldsymbol {X}=(X_{1},\dots ,X_{d})$ follows a standard GPD. From this result we can establish in Eq. 2.1 the exact distribution of the number N_x of exceedances. For $\boldsymbol {x}=(x_,\dots ,x)$ with constant entry x, we obtain in Eq. 2.3 the corresponding FI.

In Section 3 we extend the results of Section 2 to a random vector X, whose copula is in a proper neighborhood of a shifted GPD, see Eq. 3.1. The main result is Theorem 3.1. It leads to the representation (3.5) of the probability that X_i > x_i for at least k components under the mild condition that that the copula corresponding to the random vector $\boldsymbol {X}=(X_{1},\dots ,X_{d})$ is in the max-domain of attraction of a multivariate max-stable df G. Note that this condition does not require an absolutely continuously distributed random vector X.

2 Main Result for Standard GPD

The following result reveals the particular significance of ordered D-norms concerning multivariate exceedances of standard GPD.

Theorem 2.1.

Suppose the random vector $\boldsymbol {X}=(X_{1},\dots ,X_{d})$ follows a standard GPD with corresponding D-norm $\left \Vert \cdot \right \Vert _{D}$ on $\mathbb {R}^{d}$. Then we have

$$ \begin{array}{@{}rcl@{}} &&P(X_{i} > x_{i} \mathrm{for at least} k \mathrm{of the \ components} 1\le i\le d)\\ &=& P\left( N_{\boldsymbol{x}}\ge k\right)\\ &=& \left\Vert\boldsymbol{x}\right\Vert_{D,(d-k+1)}, \end{array} $$

for any $\boldsymbol {x}=(x_{1},\dots ,x_{d})$ in a left neighborhood of $\boldsymbol {0}\in \mathbb {R}^{d}$.

The assertion is obvious for k = d, because $\left \Vert \cdot \right \Vert _{D,(1)}=\wr \!\!\wr \cdot \wr \!\!\wr _{D}$. For k = 1 we obtain

$$ \begin{array}{@{}rcl@{}} &&P(X_{i} > x_{i} \text{for at least one of the components} 1\le i\le d)\\ &=& 1- P(\boldsymbol{X}\le\boldsymbol{x}) =\left\Vert\boldsymbol{x}\right\Vert_{D} =\left\Vert\boldsymbol{x}\right\Vert_{D,(d)}, \end{array} $$

for x in a left neighborhood of $\boldsymbol {0}\in \mathbb {R}^{d}$.

Proof 1 (Proof of Theorem 2.1 ).

According to equation (2.14) in Falk (2019), we can suppose the representation

$$ \boldsymbol{X}=-U\left( \frac 1{Z_{1}},\dots,\frac 1{Z_{d}} \right), $$

where the random variable U is on (0,1) uniformly distributed and independent of Z. We can also suppose by Theorem 1.7.1 in Falk (2019) that the generator Z is bounded.

Conditioning on U = u, we obtain for $\boldsymbol {x}\le \boldsymbol {0}\in \mathbb {R}^{d}$ close enough to 0, by the boundedness of Z,

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!&&P(X_{i} > x_{i} \text{for at least} k \text{of the components} \ 1\le i\le d)\\ \!\!\!\!\!\!\!&=& P(U < \left\vert x_{i}\right\vert Z_{i} \text{for at least} k \text{of the components} 1\le i\le d)\\ \!\!\!\!\!\!\!&=&{{\int}_{0}^{1}} P(u < \left\vert x_{i}\right\vert Z_{i} \text{for at least} k \text{of the components} 1\le i\le d) du\\ \!\!\!\!\!\!\!&=&{{\int}_{0}^{1}} P\big((d-k+1)\text{-th smallest value among} \left\{\left\vert x_{1}\right\vert Z_{1},\dots,\left\vert x_{d}\right\vert Z_{d}\right\}\ge u\big) du\\ \!\!\!\!\!\!\!&=&{\int}_{0}^{\infty} P\big((d - k + 1)\text{-th smallest value among} \left\{\left\vert x_{1}\right\vert Z_{1},\dots,\left\vert x_{d}\right\vert Z_{d}\right\}\ge u\big) du\\ \!\!\!\!\!\!\!&=&E((d-k+1)\text{-th smallest value among} \left\{\left\vert x_{1}\right\vert Z_{1},\dots,\left\vert x_{d}\right\vert Z_{d}\right\})\\ \!\!\!\!\!\!\!&=& \left\Vert\boldsymbol{x}\right\Vert_{D,(d-k+1)}, \end{array} $$

using the general equation $E(Y)={\int \limits }_{0}^{\infty } P(Y > u) du$ with a random variable Y ≥ 0. □

We obtain from the previous result the exact distribution of the number of exceedances for x in a left neighborhood of $\boldsymbol {0}\in \mathbb {R}^{d}$ if $\boldsymbol {X}=(X_{1},\dots ,X_{d})$ follows a standard GPD:

$$ \begin{array}{@{}rcl@{}} P(N_{\boldsymbol{x}}=k)&=&P(N_{\boldsymbol{x}}\ge k)- P(N_{\boldsymbol{x}}\ge k-1)\\ &=&\begin{cases} \left\Vert\boldsymbol{x}\right\Vert_{D,(d-k+1)}- \left\Vert\boldsymbol{x}\right\Vert_{D,(d-k)},& k=1,\dots,d-1,\\ \left\Vert\boldsymbol{x}\right\Vert_{D,(1)}= \wr\!\!\wr\boldsymbol{x}\wr\!\!\wr_{D},&k=d, \end{cases}\\ &=:& p_{\boldsymbol{x},D}(k). \end{array} $$

(2.1)

With

$$ p_{\boldsymbol{x},D}(0):= P(N_{\boldsymbol{x}}=0)= P(X_{i}\le x_{i}\text{ for }1\le i\le d) = 1-\left\Vert\boldsymbol{x}\right\Vert_{D}, $$

the numbers p_x,D(k), $k=0,\dots ,d$, define the distribution of the exceedance counts N_x on $\left \{0,1,\dots ,d\right \}$.

In case of a vector $\boldsymbol {x}=(x,\dots ,x)\in \mathbb {R}^{d}$ with constant entries, we obtain, with Z_0:d := 0,

$$ \begin{array}{@{}rcl@{}} p_{x,D}(k)&:=& p_{(x,\dots,x),D}(k)\\ &=&\begin{cases} \left\vert x \right\vert\big(E(Z_{d-k+1:d})-E(Z_{d-k:d}) \big),& 1\le k\le d,\\ 1- \left\vert x \right\vert E(Z_{d:d}),& k=0, \end{cases}. \end{array} $$

(2.2)

This reveals a crucial role of the spacingsZ_d−k+ 1:d − Z_k−d:d in the analysis of multivariate exceedances.

The expectation of N_x is by Eq. 2.2

$$ \begin{array}{@{}rcl@{}} E(N_{x})&=& \sum\limits_{k=1}^{d} kp_{x,D}(k)\\ &=& \left\vert x\right\vert \sum\limits_{k=1}^{d} k \big(E(Z_{d-k+1:d})-E(Z_{d-k:d}) \big)\\ &=& \left\vert x\right\vert \sum\limits_{k=1}^{d} E(Z_{d-(k-1):d})\\ &=& \left\vert x\right\vert E\left( \sum\limits_{k=1}^{d}Z_{d-(k-1):d}\right)\\ &=& \left\vert x\right\vert E\left( \sum\limits_{k=1}^{d}Z_{k}\right)= d \left\vert x\right\vert \end{array} $$

by the fact that E(Z_k) = 1, 1 ≤ k ≤ d. The fragility index is, therefore, with x < 0 close enough to zero,

$$ \begin{array}{@{}rcl@{}} \text{FI} &=& E(N_{x}\mid N_{x} >0)\\ &=& \frac{E(N_{x})}{1-P(N_{x}=0)}\\ &=& \frac{d\left\vert x\right\vert}{1-p_{x,D}(0)}\\ &=&\frac d{E\left( Z_{d:d}\right)} = \frac d{\left\Vert\boldsymbol{1}\right\Vert_{D}}. \end{array} $$

(2.3)

Note that in this case of a standard GPD, the number E(N_x∣N_x > 0) does not depend on x, i.e., the FI does not require a limit. The number $\left \Vert \boldsymbol {1}\right \Vert _{D}$ is known as the extremal coefficient, introduced by Smith (1990). It measures the tail dependence of the components $X_{1},\dots ,X_{d}$ by just one number. If we have $\left \Vert {\boldsymbol {1}}\right \Vert _{D}=d$, then there is tail independence, and in case $\left \Vert {\boldsymbol {1}}\right \Vert _{D}=1$ we have complete dependence, see Falk (2019), equation (2.28).

Example 2.1.

Take, for example, the Dirichlet D-norm $\left \Vert \cdot \right \Vert _{D(\alpha )}$ with parameter α = 1, see Example 1.7.4 in Falk (2019). Its generator is $\boldsymbol {Z}=(Z_{1},\dots ,Z_{d})$, where $Z_{1},\dots ,Z_{d}$ are independent and identically standard exponential distributed random variable. We have

$$ E\left( Z_{k:d}\right) = \sum\limits_{j=d-k+1}^{d} \frac 1 j,\qquad 1\le k\le d, $$

see equation (1.7.7) in Reiss (1989) and, thus, we obtain for a vector with constant entry $\boldsymbol {x}=(x,\dots ,x)\in \mathbb {R}^{d}$

$$ p_{x,D}(k)=\begin{cases} \frac{\left\vert x\right\vert}{k},& 1\le k\le d\\ 1-\left\vert x\right\vert{\sum}_{k=1}^{d}\frac 1 k,&k=0. \end{cases} $$

Recall that the previous considerations require that the vector x is in a left neighborhood of $\boldsymbol {0}\in \mathbb {R}^{d}$. Otherwise, with arbitrary x≠ 0, the preceding equation could not be true.

The fragility index is consequently,

$$ \text{FI} = \frac d{E(Z_{d:d})} = \frac d{{\sum}_{k=1}^{d} \frac 1k} $$

with the extremal coefficient

$$ \left\Vert\boldsymbol{1}\right\Vert_{D} = \sum\limits_{k=1}^{d} \frac 1k. $$

Example 2.2.

Let $U_{1},\dots ,U_{d}$ be independent and on (0,1) uniformly distributed random variables, i.e., P(U_i ≤ u) = u, u ∈ [0,1], 1 ≤ i ≤ d. Then $\boldsymbol {Z}=(Z_{1},\dots ,Z_{d}):=2(U_{1},\dots ,U_{d})$ is a generator of a D-norm $\left \Vert \cdot \right \Vert _{D}$. From the fact that E(U_k:d) = k/(d + 1), 1 ≤ k ≤ d, see equation (1.7.3) in Reiss (1989), we obtain E(Z_k:d) = 2k/(d + 1) and, thus, by Eq. 2.2,

$$ p_{x,D}(k) = \begin{cases} \frac{2\left\vert x\right\vert}{d+1},&1\le k \le d,\\ 1- 2\left\vert x\right\vert \frac d{d+1},&k=0. \end{cases} $$

In this case, the conditional probability that we have exactly k exceedances above x among $X_{1},\dots ,X_{d}$, given that there is at least one, is by Eq. 2.1,

$$ \begin{array}{@{}rcl@{}} P(N_{x}=k\mid N_{x}\ge 1) &=& \frac{P(N_{x}=k)}{P(N_{x}\ge 1)}\\ &=& \frac{p_{x,D}(k)}{1-p_{x,D}(0)} = \frac 1 d,\qquad 1\le k\le d, \end{array} $$

which is the uniform distribution on the set of integers $\left \{1,\dots ,d\right \}$. The fragility index is FI = (d + 1)/2 and the extremal coefficient is $\left \Vert \boldsymbol {1}\right \Vert _{D}=1/E(Z_{d:d}) = (d+1)/2d\to _{d\to \infty }1/2$.

Suppose that the copula C of the random vector $\boldsymbol {Y}=(Y_{1},\dots ,Y_{d})$ is a generalized Pareto copula (GPC), i.e., there exists a D-norm $\left \Vert \cdot \right \Vert _{D}$ on $\mathbb {R}^{d}$ such that

$$ C(\boldsymbol{u}) = 1-\left\Vert\boldsymbol{1}-\boldsymbol{u}\right\Vert_{D},\qquad \boldsymbol{u}_{0}\le\boldsymbol{u}\le\boldsymbol{1}\in\mathbb{R}^{d}, $$

for some $\boldsymbol {u}_{0}<\boldsymbol {1}\in \mathbb {R}^{d}$, see Section 3.1 in Falk (2019). Then we have equality in distribution

$$ \boldsymbol{Y}=_{D} \left( F_{1}^{-1}(U_{1}),\dots,F_{d}^{-1}(U_{d})\right), $$

where F_i is the df of Y_i, 1 ≤ i ≤ d, and the random vector $\boldsymbol {U}=(U_{1},\dots ,U_{d})$ follows the GPC C. Note that the random vector X := U −1 then follows a GPD with D-norm $\left \Vert \cdot \right \Vert _{D}$.

From the general equivalence F^− 1(q) > t ⇔ q > F(t), valid for q ∈ (0,1), $t\in \mathbb {R}$ and an arbitrary univariate df F, we obtain

$$ \begin{array}{@{}rcl@{}} &&P(Y_{i}>y_{i} \text{for at least} k \text{of the components} 1\le i\le d)\\ &=& P(U_{i}>F_{i}(y_{i}) \text{for at least} k \text{of the components} 1\le i\le d)\\ &=& P(U_{i}-1>F_{i}(y_{i})-1 \text{for at least} k \text{of the components} 1\le i\le d)\\ &=& P(X_{i}>F_{i}(y_{i})-1 \text{for at least} k \text{of the components} 1\le i\le d)\\ &=& \left\Vert\left( 1-F_{1}(y_{1}),\dots,1-F_{d}(y_{d})\right)\right\Vert_{D,(d-k+1)}, \end{array} $$

for y large enough.

3 Extension to Multivariate Max-Domain of Attraction

In the next step we extend the considerations in the previous section to a copula C, which satisfies the max-domain of attraction condition (1.1). In this case there exists a D-norm $\left \Vert \cdot \right \Vert _{D}$ on $\mathbb {R}^{d}$ with

$$ \underset{t\downarrow 0}{\lim} \frac{1-C(1+t\boldsymbol{x})}t=\left\Vert\boldsymbol{x}\right\Vert_{D},\qquad \boldsymbol{x}\le\boldsymbol{0}\in\mathbb{R}^{d}, $$

(3.1)

and the EVD G is given by

$$ G(\boldsymbol{x})=\exp\left( -\left\Vert\boldsymbol{x}\right\Vert_{D}\right),\qquad \boldsymbol{x}\le\boldsymbol{0}\in\mathbb{R}^{d}, $$

see Corollary 3.1.6 in Falk (2019).

Example 3.1.

Take an arbitrary Archimedean copula on $\mathbb {R}^{d}$

$$ C_{\varphi}(\boldsymbol{u}) = \varphi^{-1}\big(\varphi(u_{1})+\dots+\varphi(u_{d})\big),\qquad \boldsymbol{u}=(u_{1},\dots,u_{d})\in(0,1)^{d}, $$

where φ is a continuous and strictly increasing function from (0,1] to $[0,\infty )$ with φ(1) = 0, see, for example, McNeil and Nešlehová (2009), Theorem 2.2. Suppose that

$$ p:= -\underset{s\downarrow 0}{\lim} \frac{s\varphi\prime(1-s)}{\varphi(1-s)} $$

exists in $[1,\infty ]$. Then C_φ satisfies condition (3.1) with the D-norm $\left \Vert \cdot \right \Vert _{D}$ being the logistic one $\left \Vert {\boldsymbol {x}}\right \Vert _{D}=\left \Vert {\boldsymbol {x}}\right \Vert _{p}=\left ({\sum }_{i=1}^{p}\left \vert {x_{i}}\right \vert ^{p}\right )^{1/p}$, with parameter $p\in [1,\infty ]$ and the convention $\left \Vert {\boldsymbol {x}}\right \Vert _{\infty }=\max \limits _{1\le i\le d}\left \vert {x_{i}}\right \vert$, see Corollary 3.1.15 in Falk (2019).

Theorem 3.1.

Suppose that the random vector $\boldsymbol {U}=(U_{1},\dots ,U_{d})$ follows a copula C, which satisfies Eq. 3.1. Then we have, for each $k=1,\dots ,d$,

$$ \begin{array}{@{}rcl@{}} &&P\big(U_{i}>1+tx_{i} \mathrm{for at least} k \mathrm{of the components} 1\le i\le d\big)\\ &=& t \left\Vert\boldsymbol{x}\right\Vert_{D,(d-k+1)} + o(t) \end{array} $$

as t ↓ 0, for each $\boldsymbol {x}=(x_{1},\dots ,x_{d})\le \boldsymbol {0}\in \mathbb {R}^{d}$.

The preceding result implies for a random vector U, whose copula satisfies Eq. 3.1,

$$ \begin{array}{@{}rcl@{}} &&P\big(U_{i}>1+x_{i} \text{for at least} k \text{of the components} 1\le i\le d\big)\\ &=&\left\Vert\boldsymbol{x}\right\Vert \left\Vert\frac{\boldsymbol{x}}{\left\Vert{\boldsymbol{x}}\right\Vert}\right\Vert_{D,(d-k+1)} + o({\left\Vert\boldsymbol{x}\right\Vert}), \end{array} $$

(3.2)

$\boldsymbol {x}\le \boldsymbol {0}\in \mathbb {R}^{d}$, x≠0, as $\left \Vert \boldsymbol {x}\right \Vert \to 0$, with an arbitrary norm $\left \Vert \cdot \right \Vert$ on $\mathbb {R}^{d}$.

By repeating the arguments in Section 2, we obtain for the FI corresponding to $N_{tx}={\sum }_{i=1}^{d} 1_{(1+tx,1]}(U_{i})$, x < 0, if the copula C of $\boldsymbol {U}=(U_{1},\dots ,U_{d})$ satisfies condition (3.1),

$$ \text{FI} = \underset{t\downarrow 0}{\lim} \frac{E(N_{tx})}{1-P(N_{tx}=0)} = \frac{d}{\left\Vert\boldsymbol{1}\right\Vert_{d}}. $$

(3.3)

This was already observed in Falk and Tichy (2012), Theorem 4.1.

The following representation of $\left \Vert \boldsymbol {x}\right \Vert _{D,(d-k+1)}$ will be a crucial tool in the derivation of Theorem 3.1. We briefly explain its notation. By $\boldsymbol {e}_{i}=(0,\dots ,0,1,0,\dots ,0)\in \mathbb {R}^{d}$ we denote the i-th unit vector in $\mathbb {R}^{d}$, $i=1,\dots ,d$. Any $\boldsymbol {x}=(x_{1},\dots ,x_{d})\in \mathbb {R}^{d}$ can be represented as $\boldsymbol {x}={\sum }_{i=1}^{d} x_{i}\boldsymbol {e}_{i}$. Choose a subset $A\subset {\left \{{1,\dots ,d}\right \}}$. Then we set

$$ \boldsymbol{x}_{A}:=\sum\limits_{i\in A} x_{i}\boldsymbol{e}_{i} \in\mathbb{R}^{d}, $$

with the convention $\boldsymbol {x}_{\emptyset }=\boldsymbol {0}\in \mathbb {R}^{d}$. By $\left \vert A\right \vert$ we denote the number of elements in a set A.

Lemma 3.1.

We have for any D-norm $\left \Vert \cdot \right \Vert _{D}$ on $\mathbb {R}^{d}$ and each $k=1,\dots ,d$

$$ \begin{array}{@{}rcl@{}} & &\left\Vert\boldsymbol{x}\right\Vert_{D,(d-k+1)} \\ &=& \sum\limits_{m=k}^{d} \sum\limits_{T\subset\left\{1,\dots,d\right\}, \left\vert T\right\vert=m} \sum\limits_{S\subset T}(-1)^{\left\vert S\right\vert-1} \left\Vert\boldsymbol{x}_{S\cup T^{\complement}}\right\Vert_{D},\quad \boldsymbol{x}\in \mathbb{R}^{d}. \end{array} $$

(3.4)

The preceding probabilistic result entails the following non probabilistic representation of the (d − k + 1)-th smallest value among arbitrary nonnegative numbers $x_{1},\dots ,x_{d}$ in terms of maxima of subsets of $\left \{{x_{1},\dots ,x_{d}}\right \}$.

Choosing the particular D-norm $\left \Vert \cdot \right \Vert _{D}=\left \Vert \cdot \right \Vert _{\infty }$, with constant generator $\boldsymbol {Z}=(1,\dots ,1)$, Lemma 3.1 implies, with $\boldsymbol {x}=(x_{1},\dots ,x_{d})\ge \boldsymbol {0}\in \mathbb {R}^{d}$,

$$ \begin{array}{@{}rcl@{}} &&\text{the} (d-k+1)\text{-th smallest value among} x_{1},\dots,x_{d}\\ &=& \sum\limits_{m=k}^{d} \sum\limits_{T\subset\left\{1,\dots,d\right\}, \left\vert T\right\vert=m} \sum\limits_{S\subset T}(-1)^{\left\vert S\right\vert-1} \underset{S\cup T^{\complement}}{\max} x_{i},\qquad k=1,\dots,d. \end{array} $$

In particular for k = d we obtain

$$ \underset{1\le i\le d}{\min} x_{i} = \sum\limits_{S\subset\left\{1,\dots,d\right\}}(-1)^{\left\vert S\right\vert - 1}\underset{i\in S}{\max} x_{i}, $$

which is a well known representation of a minimum of nonnegative numbers in terms of maxima, c.f. Lemma 1.6.1 in Falk (2019).

Proof 2 (Proof of Lemma 3.1 ).

We present a probabilistic proof of Lemma 3.1. Let $\boldsymbol {X}=(X_{1},\dots ,X_{d})$ follow a standard GPD with D-norm $\left \Vert \cdot \right \Vert _{D}$, i.e.,

$$ P(\boldsymbol{X}\le \boldsymbol{x})= 1- \left\Vert\boldsymbol{x}\right\Vert_{D} $$

for all x ∈ [ε,0]^d, with some ε < 0. For such $\boldsymbol {x}=(x_{1},\dots ,x_{d})$ we obtain from Theorem 2.1, with $k\in \left \{{1,\dots ,d}\right \},$

$$ \begin{array}{@{}rcl@{}} \left\Vert\boldsymbol{x}\right\Vert_{D,(d-k+1)}&=& P\left( \sum\limits_{i=1}^{d} 1_{(x_{i},0]}(X_{i})\ge k\right)\\ &=& \sum\limits_{m=k}^{d} P\left( \sum\limits_{i=1}^{d} 1_{(x_{i},0]}(X_{i})= m\right)\\ &=& \sum\limits_{m=1}^{d} \sum\limits_{T\subset\left\{1,\dots,d\right\} \left\vert T\right\vert=m} P\left( X_{i}>x_{i}, i \in T; X_{j}\le x_{j},j\in T^{\complement}\right). \end{array} $$

The Inclusion-Exclusion Principle (see Corollary 1.6.2 in Falk (2019)) implies

$$ \begin{array}{@{}rcl@{}} && P\left( X_{i}>x_{i}, i \in T; X_{j}\le x_{j},j\in T^{\complement}\right)\\ &=& P\left( X_{i}>x_{i}, i \in T \mid X_{j}\le x_{j},j\in T^{\complement}\right) P\left( X_{j}\le x_{j},j\in T^{\complement} \right)\\ &=& \left( 1- P\left( \bigcup_{i\in T}\left\{X_{i}\le x_{i}\right\} \Big | X_{j}\le x_{j},j\in T^{\complement} \right)\right) P\left( X_{j}\le x_{j},j\in T^{\complement} \right)\\ &=& \left( 1 - \sum\limits_{\emptyset\not=S\subset T}(-1)^{\left\vert S\right\vert-1} P\left( X_{i}\le x_{i},i\in S\mid X_{j}\le x_{j}, j\in T^{\complement}\right) \right)\\ && \times P\left( X_{j}\le x_{j},j\in T^{\complement} \right)\\ &=& P\!\left( X_{j}\!\le\! x_{j},j\!\in\! T^{\complement} \right) - \sum\limits_{\emptyset\not=S\subset T}(-1)^{\left\vert S\right\vert-1} P\!\left( \!X_{i}\!\le\! x_{i},i\in S; X_{j}\!\le\! x_{j}, j\in T^{\complement}\right)\\ &=& \sum\limits_{S\subset T}(-1)^{\left\vert S\right\vert} P\left( X_{i}\le x_{i},i\in S; X_{j}\le x_{j}, j\in T^{\complement}\right)\\ &=& \sum\limits_{S\subset T}(-1)^{\left\vert S\right\vert}\left( 1- \left\Vert\boldsymbol{x}_{S\cup T^{\complement}}\right\Vert_{D} \right)\\ &=& \sum\limits_{S\subset T}(-1)^{\left\vert S\right\vert-1}\left\Vert\boldsymbol{x}_{S\cup T^{\complement}}\right\Vert_{D}, \end{array} $$

where the final equation is due to the fact that ${\sum }_{S\subset T}(-1)^{\left \vert S\right \vert } = 0$, see equation (1.10) in Falk (2019).

Altogether we have shown that, for x ∈ [ε,0]^d,

$$ \begin{array}{@{}rcl@{}} && \left\Vert\boldsymbol{x}\right\Vert_{D,(d-k+1)} \\ &=& \sum\limits_{m=k}^{d} \sum\limits_{T\subset\left\{1,\dots,d\right\}, \left\vert T\right\vert=m} \sum\limits_{S\subset T}(-1)^{\left\vert S\right\vert-1} \left\Vert\boldsymbol{x}_{S\cup T^{\complement}}\right\Vert_{D},\qquad \boldsymbol{x}\in \mathbb{R}^{d}. \end{array} $$

The fact that $\left \Vert t\boldsymbol {x}\right \Vert _{D,(d-k+1)}=t\left \Vert \boldsymbol {x}\right \Vert _{D,(d-k+1)}$ and $\left \Vert t\boldsymbol {x}_{S\cup T^{\complement }}\right \Vert _{D} = t \left \Vert \boldsymbol {x}_{S\cup T^{\complement }}\right \Vert _{D}$ for t ≥ 0 implies that the above equation is true for each $\boldsymbol {x}\in \mathbb {R}^{d}$. □

Now we can prove Theorem 3.1 in a straightforward way.

Proof 3 (Proof of Theorem 3.1 ).

Let the random vector $\boldsymbol {U}=(U_{1},\dots ,U_{d})$ follow a copula C, which satisfies Eq. 3.1. Choose $k\in \left \{{1,\dots ,d}\right \}$ and $\boldsymbol {x}=(x_{1},\dots ,x_{d})\le \boldsymbol {0}\in \mathbb {R}^{d}$. By repeating arguments in the proof of Lemma 3.1 we obtain

$$ \begin{array}{@{}rcl@{}} &&P\left( \sum\limits_{i=1}^{d} 1_{(1+tx_{i},1]}(U_{i}) \ge k\right)\\ &=& \sum\limits_{m=k}^{d} P\left( \sum\limits_{i=1}^{d} 1_{(1+tx_{i},1]}(U_{i}) =m\right)\\ &=& \sum\limits_{m=k}^{d} \sum\limits_{T\subset\left\{1,\dots,d\right\},\left\vert T \right\vert = m} P\left( U_{i}>1+tx_{i},i\in T; U_{j}\le 1+ t x_{j}, j\in T^{\complement}\right)\\ &=& \sum\limits_{m=k}^{d} \sum\limits_{T\subset\left\{1,\dots,d\right\},\left\vert T \right\vert = m} \Big\{ P\left( U_{j}\le 1+ tx_{j}, j\in T^{\complement}\right)\\ && - \sum\limits_{\emptyset\not= S\subset T} (-1)^{\left\vert S \right\vert - 1} P\left( U_{i}\le 1+t x_{i}, i\in S; U_{j}\le 1+t x_{j},j\in T^{\complement}\right)\Big\}\\ &=& \sum\limits_{m=k}^{d} \sum\limits_{T\subset\left\{1,\dots,d\right\},\left\vert T \right\vert = m} \Big\{ 1- t\left\Vert\boldsymbol{x}_{T^{\complement}}\right\Vert_{D}\\ && - \sum\limits_{\emptyset\not= S\subset T} (-1)^{\left\vert S \right\vert - 1} \left( 1- t \left\Vert\boldsymbol{x}_{S\cup T^{\complement}}\right\Vert_{D}\right)\Big\} + o(t)\\ &=& \sum\limits_{m=k}^{d} \sum\limits_{T\subset\left\{1,\dots,d\right\},\left\vert T \right\vert = m} \Big\{ 1- \sum\limits_{\emptyset\not= S\subset T} (-1)^{\left\vert S \right\vert - 1}\\ && + t \sum\limits_{S\subset T} (-1)^{\left\vert S \right\vert - 1} \left\Vert\boldsymbol{x}_{S\cup T^{\complement}}\right\Vert_{D}\Big\} + o(t)\\ &=& t \sum\limits_{m=k}^{d} \sum\limits_{T\subset\left\{1,\dots,d\right\},\left\vert T \right\vert = m} \sum\limits_{S\subset T} (-1)^{\left\vert S \right\vert - 1} \left\Vert\boldsymbol{x}_{S\cup T^{\complement}}\right\Vert_{D}\Big\} + o(t)\\ &=& t \left\Vert\boldsymbol{x}\right\Vert_{D,(d-k+1)} + o(t) \end{array} $$

by Lemma 3.1. This completes the proof of Theorem 3.1. □

Consider next a random vector $\boldsymbol {X}=(X_{1},\dots ,X_{d})$ that is in the max-domain of attraction of a multivariate max-stable df G. This is equivalent with the condition that the copula C corresponding to X satisfies Eq. 3.1, together with the condition that, for each $i=1,\dots ,d$, the (univariate) df F_i of X_i is in the max-domain of attraction of a univariate max-stable df G_i; see, e.g. Proposition 3.1.10 in Falk (2019).

Then we obtain from Eq. 3.2, with $\boldsymbol {U}=(U_{1},\dots ,U_{d})$ following the copula C, so that $\boldsymbol {X}=(X_{1},\dots ,X_{d})=_{\mathcal D}\left (F_{1}^{-1}(U_{1}),\dots ,F_{d}^{-1}(U_{d})\right )$

$$ \begin{array}{@{}rcl@{}} &&P\left( X_{i}>y_{i} \text{for at least} k \text{of the components} i=1,\dots,d \right)\\ &=& P\left( U_{i}> 1+(F_{i}(y_{i})-1) \text{for at least} k \text{of the components} i=1,\dots,d \right)\\ &=& \left\Vert\left( 1-F_{i}(y_{i})\right)_{i=1}^{d}\right\Vert \times \left\Vert\frac{\left( 1-F_{i}(y_{i})\right)_{i=1}^{d}}{\left\Vert{\left( 1-F_{i}(y_{i})\right)_{i=1}^{d}}\right\Vert}\right\Vert_{D,(d-k+1)}\\ &&+ o\left( \left\Vert\left( 1-F_{i}(y_{i})\right)_{i=1}^{d}\right\Vert\right),\qquad \boldsymbol{y}=(y_{1},\dots,y_{d})\in\mathbb{R}^{d}, \end{array} $$

(3.5)

with an arbitrary norm $\left \Vert \cdot \right \Vert$ on $\mathbb {R}^{d}$. Note that we actually do not have to require in Eq. 3.5 that each univariate margin F_i is in the max-domain of attraction of a univariate max-stable df. The preceding equation is, therefore, true if the copula C corresponding to X satisfies condition (3.1). Note that this condition on the copula of X does not require that X is absolutely continuous, i.e., Eq. 3.5 is true for arbitrary univariate df $F_{1},\dots ,F_{d}$, not necesscarily continuous ones.

But, if each F_i is in the max-domain of attraction of a max-stable df G_i, then there exist constants a_it > 0, $b_{it}\in \mathbb {R}$ for t > 0, with

$$ t \big(1-F_{i}(a_{it}y+b_{it})\big) \to_{t\to\infty} -\log(G_{i}(y)),\qquad y\in\mathbb{R}; $$

see, for example, Falk (2019), equation (2.3). As a consequence we obtain from Eq. 3.5

$$ \begin{array}{@{}rcl@{}} &&P\left( X_{i}>a_{it}y_{i}+b_{it} \text{for at least} k \text{of the components} i=1,\dots,d \right)\\ &=& \frac 1t \left\Vert t \left( 1-F_{i}(a_{it}y_{i} + b_{it})\right)_{i=1}^{d}\right\Vert_{D,(d-k+1)} + o\left( \frac 1 t\right) \\ &=&\frac 1 t \left\Vert\big (\log(G_{i}(y_{i}))\big)_{i=1}^{d} \right\Vert_{D,(d-k+1)} + o\left( \frac 1 t\right) \end{array} $$

if G_i(y_i) ∈ (0,1], 1 ≤ i ≤ d.

Suppose identical distributions of the components of X, i.e., $F_{1}=\dots =F_{d}=: F$ and identical entries of y, i.e., $y_{1}=\dots =y_{d}=: y$. Then we can repeat the arguments in Section 2, with $\boldsymbol {x}:=(F(y)-1,\dots ,F(y)-1)$, and obtain, with $N_{y}:={\sum }_{i=1}^{d} 1_{(y,\infty )}(X_{i})$,

$$ \begin{array}{@{}rcl@{}} p_{y,D}(k)&=& P(N_{y}=k)\\ &=& \begin{cases} (1-F(y))\big(E(Z_{d-k+1:d})- E(Z_{d-k:d}) \big),&1\le k\le d,\\ 1-(1-F(y))E(Z_{d:d}) \end{cases}\\ &&+ o(1-F(y)) \end{array} $$

as $y\uparrow \omega (F):= \sup \left \{t\in \mathbb {R}: F(t)<1\right \}$.

In particular we obtain for the FI

$$ \begin{array}{@{}rcl@{}} \text{FI}&=& \underset{y\uparrow \omega(F)}{\lim} E(N_{y}\mid N_{y}>0)\\ &=& \frac{E(N_{y})}{1-P(N_{y}=0)} = \frac d{\left\Vert\boldsymbol{1}\right\Vert_{D}}, \end{array} $$

as already observed in (Falk and Tichy, 2012, Theorem 5.1).

If, for example, the underlying D-norm is a logistic one, $\left \Vert \boldsymbol {x}\right \Vert _{D}=\left \Vert \boldsymbol {x}\right \Vert _{p}:=\left ({\sum }_{i=1}^{d}\left \vert x_{i}\right \vert ^{p}\right )^{1/p}$, with parameter $p\in [1,\infty ]$, and the convention $\left \Vert {\boldsymbol {x}}\right \Vert _{\infty }=\max \limits _{1\le i\le d}\left \vert {x_{i}}\right \vert$, then the FI is

$$ \text{FI} = \frac d{\left\Vert\boldsymbol{1}\right\Vert_{p}}= \begin{cases} 1,&\text{if }p=1,\\ d^{1-\frac 1p}, &\text{if }p\in(1,\infty)\\ d,&\text{if }p=\infty. \end{cases} $$

If p = 1, the components $X_{1},\dots ,X_{d}$ are tail independent with FI = 1, i.e., the system $\left \{X_{1},\dots ,X_{d}\right \}$ is stable. If $p=\infty$, the components $X_{1},\dots ,X_{d}$ are completely tail dependent and FI = d, i.e., the system is extremely fragile.