Estimation of extreme Component-wise Excess design realization: a hydrological application

Abstract

The classic univariate risk measure in environmental sciences is the Return Period (RP). The RP is traditionally defined as “the average time elapsing between two successive realizations of a prescribed event”. The notion of design quantile related with RP is also of great importance. The design quantile represents the “value of the variable(s) characterizing the event associated with a given RP”. Since an individual risk may strongly be affected by the degree of dependence amongst all risks, the need for the provision of multivariate design quantiles has gained ground. In contrast to the univariate case, the design quantile definition in the multivariate setting presents certain difficulties. In particular, Salvadori, G., De Michele, C. and Durante F. define in the paper called “On the return period and design in a multivariate framework” (Hydrol Earth Syst Sci 15:3293–3305, 2011) the design realization as the vector that maximizes a weight function given that the risk vector belongs to a given critical layer of its joint multivariate distribution function. In this paper, we provide the explicit expression of the aforementioned multivariate risk measure in the Archimedean copula setting. Furthermore, this measure is estimated by using Extreme Value Theory techniques and the asymptotic normality of the proposed estimator is studied. The performance of our estimator is evaluated on simulated data. We conclude with an application on a real hydrological data-set.

Appendices

Appendix 1: Definitions of regular varying functions

In the following, the notions of regularly varying (RV) and second-order regularly varying (2RV) function are introduced. These definitions are useful in Sect. 3. For more details, the reader is referred to Section B.1 in de Haan and Ferreira (2006) and Mao and Hu (2012).

Definition 6.1

(RV function). A measurable function, $h:{\mathbb {R}}_+\rightarrow {\mathbb {R}}$ that is eventually positive, is said to be of regular variation at infinity with index $\gamma \in {\mathbb {R}} \backslash \{0\}$, denoted by $h\in RV_\gamma (+\infty )$, if for any $x>0$,

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{h(tx)}{h(t)}=x^\gamma . \end{aligned}$$

(15)

If (15) holds with $\gamma =0$ for any $x>0$, then h is said to be slowly varying at infinity and is written as $h\in RV_0(+\infty )$.

Definition 6.2

(2RV function). A measurable function, $h:{\mathbb {R}}_+\rightarrow {\mathbb {R}}$ that is eventually positive, is said to be of second-order regular variation with the first-order parameter $\gamma \in {\mathbb {R}}$ and the second-order parameter $\tau \le 0$, denoted by $h\in 2RV_{\gamma ,\tau }(A)$, if there exists some ultimately positive or negative function A(t) with $A(t)\rightarrow 0$ as $t\rightarrow \infty$ such that

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{\frac{h(tx)}{h(t)}-x^\gamma }{A(t)}= H(x), \text{ with } H(x)=x^\gamma \int _{1}^x s^{\tau -1} ds, \forall x>0. \end{aligned}$$

Here, A is referred to as an auxiliary function of h. It is verified that $|A|\in RV_\tau (+\infty )$.

Appendix 2: First- and second-order tail behaviour of $Y\mathop {=}\limits ^{d}\max \{V_1, \ldots , V_d\}$

Let $Y\mathop {=}\limits ^{d}\max \{V_1, \ldots , V_d\}$ be the random variable introduced in Sect. 3. The (first-order) tail behaviour of Y is studied in the following proposition. The interested reader is also referred to Theorem 1.1.8 in de Haan and Ferreira (2006).

Proposition 7.1

(The von Mises condition for Y). Assume that ${{\varvec{X}}}=(X_1,\ldots , X_d)$ satisfies conditions (a) and (b) in Assumption 2.1. Assume that $\phi _{\varvec{\theta }} \in RV_{\rho }(1)$, with $\rho \in [1,+\infty )$. Let $Y\mathop {=}\limits ^{d}\max \{V_1, \ldots , V_d\}$ with $V_i\mathop {=}\limits ^{d}F_i(X_i)$, for $i \in \{1, \ldots , d\}$. Then, $F_{{{Y}}}$ verifies the von Mises condition with tail index $\gamma _{{{{Y}}}}=-1$. In particular, $Y \in MDA(\Psi _{\gamma _Y = -1})$.

Proof

From Sect. 3, we obtain

$$\begin{aligned} F'_{{{Y}}}(t)=(\phi _{\varvec{\theta }}^{-1})'[ d \, \phi _{\varvec{\theta }}(t)]\, d \, \phi _{\varvec{\theta }}'(t)\quad \hbox {and} \quad F''_{{{Y}}}(t)=(\phi _{\varvec{\theta }}^{-1})''[ d \, \phi _{\varvec{\theta }}(t)] \, d^2 \, (\phi _{\varvec{\theta }}'(t))^2 + (\phi _{\varvec{\theta }}^{-1})'[ d \, \phi _{\varvec{\theta }}(t)]\, d \, \phi _{\varvec{\theta }}''(t). \end{aligned}$$

The limit in Equation (1.1.30) in de Haan and Ferreira (2006) can now be calculated.

$$\begin{aligned} \lim _{t\uparrow 1} \frac{(1- F_{{{Y}}}(t))F_{{{Y}}}''(t)}{(F_{{{Y}}}'(t))^2}= & {} \lim _{t\uparrow 1} \frac{(1- \phi _{\varvec{\theta }}^{-1}(d \,\phi _{\varvec{\theta }}(t))) \, \left( (\phi _{\varvec{\theta }}^{-1})''[ d \, \phi _{\varvec{\theta }}(t)] \, d^2 \, (\phi _{\varvec{\theta }}'(t))^2+(\phi _{\varvec{\theta }}^{-1})'[ d \, \phi _{\varvec{\theta }}(t)] \, d \, \phi _{\varvec{\theta }}''(t)\right) }{\left( (\phi _{\varvec{\theta }}^{-1})'[ d \, \phi _{\varvec{\theta }}(t)] \, d \, \phi _{\varvec{\theta }}'(t)\right) ^2} \\= & {} \lim _{t\uparrow 1} (1- \phi _{\varvec{\theta }}^{-1}(d \,\phi _{\varvec{\theta }}(t))) \, (\phi _{\varvec{\theta }}^{-1})''[ d \, \phi _{\varvec{\theta }}(t)] \, (\phi _{\varvec{\theta }}'(\phi _{\varvec{\theta }}^{-1}(d \,\phi _{\varvec{\theta }}(t))))^2\\+ & {} \lim _{t\uparrow 1} (1- \phi _{\varvec{\theta }}^{-1}(d \, \phi _{\varvec{\theta }}(t))) \, \phi _{\varvec{\theta }}'(\phi _{\varvec{\theta }}^{-1}(d \, \phi _{\varvec{\theta }}(t)))\, \frac{\phi _{\varvec{\theta }}''(t)}{d \,(\phi _{\varvec{\theta }}'(t))^2} . \end{aligned}$$

Given the assumptions $\phi _{\varvec{\theta }}\in RV_\rho (1)$, then $\phi _{\varvec{\theta }}'\in RV_{\rho -1}(1)$. Therefore,

$$\begin{aligned} \lim _{t\uparrow 1} (1- \phi _{\varvec{\theta }}^{-1}(d \, \phi _{\varvec{\theta }}(t))) \, (\phi _{\varvec{\theta }}^{-1})''[ d \, \phi _{\varvec{\theta }}(t)]\,(\phi _{\varvec{\theta }}'(\phi _{\varvec{\theta }}^{-1}(d \,\phi _{\varvec{\theta }}(t))))^2=\rho -1 \end{aligned}$$

and

$$\begin{aligned} \lim _{t\uparrow 1} (1- \phi _{\varvec{\theta }}^{-1}(d \,\phi _{\varvec{\theta }}(t)))\,\phi _{\varvec{\theta }}'(\phi _{\varvec{\theta }}^{-1}(d \, \phi _{\varvec{\theta }}(t)))\,\frac{\phi _{\varvec{\theta }}''(t)}{d \, (\phi _{\varvec{\theta }}'(t))^2}=-\rho +1. \end{aligned}$$

Finally, we conclude $\lim _{t\uparrow 1} \frac{(1- F_{{{Y}}}(t))F_{{{Y}}}''(t)}{(F_{{{Y}}}'(t))^2}=0 \Rightarrow \gamma _{{{Y}}}=-1.$ Hence the result follows. $\square$

The (second-order) tail behaviour of Y is studied in the following proposition.

Proposition 7.2

Assume that ${{\varvec{X}}}=(X_1,\ldots , X_d)$ satisfies conditions (a), (b) and (d) in Assumption 2.1. Let $Y\mathop {=}\limits ^{d}\max \{V_1, \ldots , V_d\}$ with $V_i\mathop {=}\limits ^{d}F_i(X_i)$, for $i \in \{1, \ldots , d\}$. Then, $1-U_{{{Y}}}\in 2RV_{-1, \beta }(A_{{{Y}}})$.

Proof

From Sect. 3, it is known that

$$\begin{aligned} \overline{F}_{{{Y}}}(1-1/t)=1- \phi _{\varvec{\theta }}^{-1}\left( d \, \phi _{\varvec{\theta }}(1-1/t)\right) . \end{aligned}$$

Since $x\rightarrow \phi _{\varvec{\theta }}(1-1/x)\in 2RV_{-\rho ,\beta }(A_{{{Y}}})$, from Proposition 2.4 in Mao and Hu (2012), we can write

$$\begin{aligned} \overline{F}_{{{Y}}}(1-1/t)=1- \phi _{\varvec{\theta }}^{-1}\left[ d \, c \, t^{-\rho }\left( 1+\frac{1}{\beta }A_{{{Y}}}(t)+o(A_{{{Y}}}(t))\right) \right] . \end{aligned}$$

In addition, $x\rightarrow \phi _{\varvec{\theta }}(1-1/x)\in 2RV_{-\rho ,\beta }(A_{{{Y}}})$ implies $\phi _{\varvec{\theta }}\in RV_{\rho }(1)$. From Remark C in Di Bernardino and Rullière (2014), it is verified that $1-\phi ^{-1}_{\varvec{\theta }}(1/x)\in RV_{-1/\rho }(+\infty )$. We now obtain

$$\begin{aligned} \overline{F}_{{{Y}}}(1-1/t)=(d c)^{1/\rho } \, t^{-1}\left( 1+\frac{1}{\beta }A_{{{Y}}}(t)+o(A_{{{Y}}}(t))\right) ^{1/\rho }. \end{aligned}$$

By using Taylor expansion,

$$\begin{aligned} \overline{F}_{{{Y}}}(1-1/t)=(d c)^{1/\rho } \, t^{-1}\left( 1+\frac{1}{\beta \, \rho }A_{{{Y}}}(t)+o(A_{{{Y}}}(t))\right) . \end{aligned}$$

It can be observed that $|{\tilde{A}}_{{{Y}}}|{:}{=}|\frac{1}{\rho }A_{{{Y}}}|\in RV_{\beta }$ since assumptions. From Proposition 2.4 in Mao and Hu (2012), the result is given. $\square$

Appendix 3: Proofs

1.1 Proof of Proposition 2.1

Let $\alpha \in (0,1)$. Let C and $\overline{C}$ be, respectively, the copula and the joint survival function associated to the random vector $\mathbf V =(V_1, \ldots , V_d)$. Note that the copula version of the constrained optimization problem in Eq. (1) can be written as

$$\begin{aligned} \delta _{{CE}}(\alpha )=\underset{\mathbf{v } \,\in \, \partial L_C(\alpha )}{\arg \max }\ \overline{C}(v_1, \ldots , v_d), \end{aligned}$$

(16)

with $\partial L_{C}(\alpha )=\{\mathbf{v } \in [0,1]^d: C(\mathbf{v })=\alpha \}.$

Equivalently, one can write the constrained optimization problem (16) as

$$\begin{aligned}&\underset{\mathbf{s }}{\arg \max }\,\, \overline{C}\,(\phi _{\varvec{\theta }}^{-1}( s_1) , \ldots , \phi _{\varvec{\theta }}^{-1} (s_d)) \\&\quad \text {s.t.:} \sum _{i=1}^d s_i=\phi _{\varvec{\theta }}(\alpha ), \\&\quad s_i \ge 0, \, \, \text{ for } \,\, i = 1, \ldots , d. \end{aligned}$$

(17)

Our aim is to find the solution of the constrained optimization problem in (17). From Theorem 2.21 in Boche and Jorswieck (2007), if $\overline{C}(\phi _{\varvec{\theta }}^{-1}( s_1) , \ldots , \phi _{\varvec{\theta }}^{-1} (s_d))$ is a schur-concave function¹, the global maximum for the problem in (17) is achieved by ${{\varvec{s}}}^{*}=\left( \frac{\phi _{\varvec{\theta }}(\alpha )}{d}, \ldots , \frac{\phi _{\varvec{\theta }}(\alpha )}{d} \right)$. We now study the schur-concavity of the function $\overline{C}(\phi _{\varvec{\theta }}^{-1}( s_1) , \ldots , \phi _{\varvec{\theta }}^{-1} (s_d))$.

We now prove that the d-dimensional survival copula $\overline{C}$ associated to an Archimedean copula is a schur-concave function. To this end, it is helpful to realize that by using the symmetry property, one can take $d = 2$ without loss of generality. The interested reader is referred to Marshall et al. (2011) (Section A.5) for further details. That is, it is sufficient to prove that bivariate survival copula $\overline{C}$ is a schur-concave function. In addition, every Archimedean copula is schur-concave (see Lemma 10.2.2 in Durante (2006) and Proposition 4.11 in Dolati and Dehgan Nezhad 2014). Furthermore, from Proposition 10.1.7 in Durante (2006), a bivariate copula C is schur-concave if and only if ${\hat{C}}(u,v)$ ² associated to C is also a schur-concave function.

Since, in our case C is an Archimedean copula (see condition (a) in Assumption 2.1) then ${\hat{C}}(u,v)$ is also a schur-concave function. Remark that $\overline{C}(u,v)={\hat{C}}(1-u,1-v)$, for $(u,v)\in [0,1]^2$ (see Nelsen 2006). Then to obtain the desired result, we have to prove that ${\hat{C}}(f(u),f(v)))$, with $f(u)=1-u$, $u\in [0,1]$, is a schur-concave function. This last statement holds true because ${\hat{C}}(f(u),f(v))$ is a composition of a an increasing schur-concave function (${\hat{C}}$) and a concave function (f). The interested reader is referred to Marshall et al. (2011) (Section B.2) for further details. Using similar arguments, since $\overline{C}$ is a decreasing schur-concave function and $\phi _{\varvec{\theta }}^{-1}$ a convex function, $\overline{C}(\phi _{\varvec{\theta }}^{-1}( s_1) , \ldots , \phi _{\varvec{\theta }}^{-1} (s_d))$ is a schur-concave function.

Finally, by taking $s_i=\phi (v_i)$, for $i = 1, \ldots , d$, from Theorem 2.21 in Boche and Jorswieck (2007), the global maximum in problem (17) is achieved by ${{\varvec{v}}}^*=\left( \phi _{\varvec{\theta }}^{-1}\left( \frac{\phi _{\varvec{\theta }}(\alpha )}{d}\right) , \ldots , \phi _{\varvec{\theta }}^{-1}\left( \frac{\phi _{\varvec{\theta }}(\alpha )}{d} \right) \right)$. By using the Probability Integral Transform Theorem (see Sect. 1.5.8.3 in Denuit et al. 2005) for each margin, we obtain the result. More precisely, the global optimum point for the constrained optimization problem in Eq. (1) is given by $\left( F_1^{-1}(v_1^*), \ldots , F_d^{-1}(v_d^*)\right)$. $\square$

1.2 Proof of Theorem 3.1

Firstly, note that

$$\begin{aligned} {\widehat{v}}_n({\widehat{x}}^{{{Y}}}_{1-\alpha _n}-x^{{{Y}}}_{1-\alpha _n})\mathop {\rightarrow }\limits ^{d} \Theta _1, \end{aligned}$$

(18)

where $\Theta _1=\Gamma +B+\Lambda +\frac{\lambda }{\beta -1}$ with B a standard normal and, $\Gamma$ and $\Lambda$ normal distributions as defined in Theorems 3.6.1 and 4.3.1 in de Haan and Ferreira (2006).

Indeed, from Proposition 7.2 in “Appendix 2,” condition (d) in Assumption 2.1 implies that $1-U_{{{Y}}}\in 2ERV_{-1,\beta }(A_{{{Y}}})$ and $a(t)=(1-U_{{{Y}}}(t))$. Therefore, the asymptotic result in Eq. (18) results from Theorems 3.6.1 and 4.3.1, and Corollary 4.3.2 in de Haan and Ferreira (2006).

Consequently, we obtain

$$\begin{aligned} {\widehat{v}}_n({\hat{p}}_{n}-p_{n})\mathop {\rightarrow }\limits ^{d} - \Theta _1, \end{aligned}$$

(Theorem on page 24 in Serfling 1980).

Under conditions 3, 4 and 5 in Theorem 3.1, and by applying Theorem 4.3.8 in de Haan and Ferreira (2006), we determine that

$$\begin{aligned} \frac{\sqrt{k_i}}{\log {(d^i_n)}}\left( \frac{{\widehat{x}}^i_{p_n}}{x^i_{p_n}}-1\right) \mathop {\rightarrow }\limits ^{d} \Theta _2, \end{aligned}$$

where $\Theta _2$ is a normal random variable with mean $\lambda _i / (1-\tau _i)$ and variance $\gamma ^2_i$ (see Theorem 3.2.5 in de Haan and Ferreira 2006).

We now write ${\widehat{x}}^i_{{\hat{p}}_n}$ as a function of ${\widehat{x}}^i_{p_n}$. That is, we can write

$$\begin{aligned} {\widehat{x}}^i_{{\hat{p}}_n} = X^i_{n-k_i,n} \left( \frac{k_i}{n \, \frac{{\hat{p}}_n}{p_n} \, p_n}\right) ^{{\widehat{\gamma }}_i}= {\widehat{x}}^i_{p_n} \left( \frac{{\hat{p}}_n}{p_n}\right) ^{-{\widehat{\gamma }}_i}. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \frac{\sqrt{k_i}}{\log {(d^i_n)}}\left( \frac{{\widehat{x}}^i_{{\hat{p}}_n}}{x^i_{p_n}}-1\right)= & {} \frac{\sqrt{k_i}}{\log {(d^i_n)}}\left( \frac{{\hat{p}}_n}{p_n}\right) ^{-{\widehat{\gamma }}_i}\left( \frac{{\widehat{x}}^i_{p_n}}{x^i_{p_n}}-1\right) + \frac{\sqrt{k_i}}{\log {(d^i_n)}}\left[ \left( \frac{{\hat{p}}_n}{p_n}\right) ^{-{\widehat{\gamma }}_i}-1\right] . \end{aligned}$$

(19)

On the other hand,

$$\begin{aligned} \frac{{\hat{p}}_n}{p_n}=\frac{- \Theta _1}{v_n}+ 1+ o_{{\mathbb {P}}}\left( \frac{1}{v_n}\right) . \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \frac{\sqrt{k_i}}{\log {(d^i_n)}}\left[ \left( \frac{{\hat{p}}_n}{p_n}\right) ^{-{\widehat{\gamma }}_i}-1\right] = \frac{\sqrt{k_i}}{\log {d^i_n}}\left[ \left( \frac{-\Theta _1}{v_n}+ 1+ o_{{\mathbb {P}}}\left( \frac{1}{v_n}\right) \right) ^{-{\widehat{\gamma }}_i}-1\right] . \end{aligned}$$

By using Taylor expansion and condition (13), it is verified that

$$\begin{aligned} \frac{\sqrt{k_i}}{\log {(d^i_n)}}\left[ \left( \frac{{\hat{p}}_n}{p_n}\right) ^{-{\widehat{\gamma }}_i}-1\right] \mathop {\rightarrow }\limits ^{{\mathbb {P}}}0. \end{aligned}$$

(20)

In addition, Eq. (20) implies that

$$\begin{aligned} \left( \frac{{\hat{p}}_n}{p_n}\right) ^{-{\widehat{\gamma }}_i}\mathop {\rightarrow }\limits ^{{\mathbb {P}}}1. \end{aligned}$$

(21)

Finally, by using (20) and (21) in Eq. (19), from Slutsky’s Theorem, we attain the result. $\square$