Conditional normal extreme-value copulas

Abstract

We propose a new class of extreme-value copulas which are extreme-value limits of conditional normal models. Conditional normal models are generalizations of conditional independence models, where the dependence among observed variables is modeled using one unobserved factor. Conditional on this factor, the distribution of these variables is given by the Gaussian copula. This structure allows one to build flexible and parsimonious models for data with complex dependence structures, such as data with spatial dependence or factor structure. We study the extreme-value limits of these models and show some interesting special cases of the proposed class of copulas. We develop estimation methods for the proposed models and conduct a simulation study to assess the performance of these algorithms. Finally, we apply these copula models to analyze data on monthly wind maxima and stock return minima.

Appendix

1.1 A.1 Proof of Proposition 1

With $\gamma _{j} = \frac {1}{k+1}A_{j}^{\prime }(\frac {k}{k+1}) + A_{j}(\frac {k}{k+1})$ and $\eta _{j}(k) = (k+1)A_{j}(\frac {k}{k+1}) - k$, we find that $C_{j|0}(u|u^{k}) = \gamma _{j} u^{\eta _{j}(k)}$, j = 1,…,d, and

$$ \begin{array}{@{}rcl@{}} C_{\mathbf{U}}(\mathbf{u}) &=& {\int}_{0}^{\infty} C_{N}\{C_{1|0}(u|u^{k}), \ldots, C_{d|0}(u|u^{k})\} u^{k} \ln u \mathrm{d} k \\ &\leq& {\int}_{0}^{\infty} C_{N}\{\gamma^{*} u^{\eta^{*}(k)}, \ldots, \gamma^{*} u^{\eta^{*}(k)}\} u^{k} \ln u \mathrm{d} k \\&& \sim_{u \to 0} \ {\int}_{0}^{\infty} (\gamma^{+}u)^{\kappa_{\boldsymbol{\Sigma}}\cdot\eta^{*}(k) + k} \ell_{N}(u) \mathrm{d} k, \end{array} $$

where ℓ_N(u) is a slowly varying function and $\gamma ^{*} = \max \limits _{j}\gamma _{j}, \ \eta ^{*}(k) = \min \limits _{j}\eta _{j}(k)$. It is seen that $\kappa _{L} \geq \min \limits _{k \geq 0}\{\kappa _{\boldsymbol {\Sigma }}\cdot \eta ^{*}(k) + k\} > 1$ because $\eta ^{*}(k) \geq \max \limits (0, 1 - k)$ and $\eta ^{*}(1) = \min \limits _{j} \kappa _{j} - 1 > 0$.

Similarly, one can show that $\kappa _{L} \leq \min \limits _{k \geq 0}\{\kappa _{\boldsymbol {\Sigma }}\cdot \eta ^{**}(k) + k\} \leq \kappa _{\boldsymbol {\Sigma }}\cdot \eta ^{**}(0) = \kappa _{\boldsymbol {\Sigma }}$, where $\eta ^{**}(k) = \max \limits _{j}\eta _{j}(k)$. This implies that C_U has intermediate lower tail dependence. $\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \square$

1.2 A.2 Proof of Proposition 2

Let $\bar C_{j|0}(u_{j}|u_{0}) = 1-C_{j|0}(1-u_{j}|1-u_{0})$, j = 1,…,d. We use Theorem 8.76 of Joe (2014):

$$ \begin{array}{@{}rcl@{}} \ell(w_{1},\ldots,w_{d}) &=& \lim_{u \to 0} \frac{1}{u}\left[1-{{\int}_{0}^{1}}C_{N}\{1-\bar C_{1|0}(uw_{1}|w_{0}),\ldots,1-\bar C_{d|0}(uw_{d}|w_{0}); \boldsymbol{\Sigma}\} \mathrm{d} w_{0}\right]\\ &=& \lim_{u \to 0} {\int}_{0}^{1/u}\left[1 - C_{N}\{1-\bar C_{1|0}(uw_{1}|uw_{0}),\ldots,1-\bar C_{d|0}(uw_{d}|uw_{0}); \boldsymbol{\Sigma}\} \right]\mathrm{d} w_{0}\\ &=& {\int}_{0}^{\infty}\left[1- C_{N}\{1-b_{1|0}(w_{1}|w_{0}),\ldots,1-b_{d|0}(w_{d}|w_{0}); \boldsymbol{\Sigma}\} \right]\mathrm{d} w_{0}. \end{array} $$

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \square$

1.3 A.3 Proof of Proposition 3

We have C_j|0(1 − uw_j|v₀) = 1 − uw_jc_j,0(1,v₀) + o(u), j = 1,…,d, and therefore

$$ \begin{array}{@{}rcl@{}} \ell(w_{1}, \ldots, w_{d}) &=& \lim_{u\to 0}\frac{1}{u}{{\int}_{0}^{1}}\left[1 - \min_{j}\{C_{j|0}(1-uw_{j}| v_{0})\}\right]\mathrm{d} v_{0}\\ &=& \lim_{u\to 0}\frac{1}{u}{{\int}_{0}^{1}}\left[uw_{j}\max_{j}\{c_{j,0}(1, v_{0}) + o(1)\}\right]\mathrm{d} v_{0} = {{\int}_{0}^{1}} \max_{j}\{w_{j}c_{j,0}(1,v_{0})\} \mathrm{d} v_{0}. \end{array} $$

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \square$

1.4 A.4 Proof of Proposition 4

We use the following Lemma to prove this proposition.

Lemma 1

Let C_N(u,v; ρ) be the normal copula with correlation ρ. If ρ → 1, then

$$ \begin{array}{@{}rcl@{}} C_{N}(u,u;\rho) &=& u - \left( \frac{1-\rho}{\pi}\right)^{1/2}\phi\{{\Phi}^{-1}(u)\} + (1-\rho)^{3/2}\phi\{{\Phi}^{-1}(u)\}[K_{1}(\rho) \\&&+ K_{2}(\rho)\{{\Phi}^{-1}(u)\}^{2}], \end{array} $$

where $\max \limits _{\rho } |K_{i}(\rho )| \leq K_{0} < \infty$, i = 1, 2.

Proof Proof of Lemma 1:

Denote C_N(u|v; ρ) = ∂C_N(u,v; ρ)/∂v and $\delta = \sqrt {1-\rho }$. We find that

$$C_{N}(u,u;\rho) = 2{{\int}_{0}^{u}} C_{N}(v|v;\rho)\mathrm{d} v = 2{{\int}_{0}^{u}}{\Phi}\left\{\frac{\delta}{\sqrt{2-\delta^{2}}}{\Phi}^{-1}(v)\right\} \mathrm{d} v.$$

Let $h(\delta ) = h(\delta , v) = {\Phi }\left \{\frac {\delta }{\sqrt {2-\delta ^{2}}}{\Phi }^{-1}(v)\right \}$ and $g(\delta ) = g(\delta , v) = \phi \left \{\frac {\delta }{\sqrt {2-\delta ^{2}}}{\Phi }^{-1}(v)\right \}$. Using the Taylor expansion of h(δ) around δ = 0 with a fixed v yields:

$$ h(\delta) = h(0) + h^{\prime}(0)\delta + h^{\prime\prime}(\varUpsilon)\delta^{2}, \quad 0 < \varUpsilon < \delta, $$

where

$$ h^{\prime}(t) = \frac{2{\Phi}^{-1}(v)}{(2-t^{2})^{1.5}} \cdot g(t), \quad h^{\prime\prime}(t) = -\frac{2t{\Phi}^{-1}(v)}{(2-t^{2})^{3.5}} \cdot [2\{{\Phi}^{-1}(v)\}^{2} - 3(2-t^{2})]g(t). $$

It implies that

$$h(\delta) = 0.5 + 0.5\pi^{-1/2}\delta {\Phi}^{-1}(v) + w_{1}\delta^{3} {\Phi}^{-1}(v) - w_{2}\delta^{3} \{{\Phi}^{-1}(v)\}^{3},$$

where 0 < w₁ < 6ϕ(0) and 0 < w₂ < 4ϕ(0) for any 0 < v < 1 and δ → 0; hence,

$$ C_{N}(u,u;\rho) = 2{{\int}_{0}^{u}} h(\delta, v) \mathrm{d} v = u + \pi^{-1/2}\delta I_{1} + 2w_{1}\delta^{3} I_{1} - 2w_{2}\delta^{3} I_{2}, $$

where

$$ I_{1} = {{\int}_{0}^{u}} {\Phi}^{-1}(v) \mathrm{d} v = -\phi\{{\Phi}^{-1}(u)\}, \quad I_{2} = {{\int}_{0}^{u}} \{{\Phi}^{-1}(v)\}^{3} \mathrm{d} v = -[2+\{{\Phi}^{-1}(u)\}^{2}]\phi\{{\Phi}^{-1}(u)\}. $$

Finally,

$$ C_{N}(u,u;\rho) = u - \pi^{-1/2}\delta \phi\{{\Phi}^{-1}(u)\} + \delta^{3}[K_{1} + K_{2}\{{\Phi}^{-1}(u)\}^{2}]\phi\{{\Phi}^{-1}(u)\}, $$

where K₁ = − 2w₁ + 4w₂, K₂ = 2w₂ and |K₁| < 24ϕ(0), K₂ < 12ϕ(0). □

Proof Proof o Proposition 4:

Note that $\phi \{{\Phi }^{-1}(u)\} \sim u \ell ^{*}(u)$ as u → 0, where ℓ^∗(u) is a slowly varying function. It implies that for any m ≥ 0,

$${\int}_{0}^{\infty} [{\Phi}^{-1}\{b_{1|0}(1|w_{0})\}]^{m}\phi[{\Phi}^{-1}\{b_{1|0}(1|w_{0})\}] \mathrm{d} w_{0} < \infty.$$

From Proposition 2 and Lemma 1, we find that, as ρ → 1,

$$ \begin{array}{@{}rcl@{}} \ell(1,1) &=& {\int}_{0}^{\infty}[1- C_{N}\{1-b_{1|0}(1|w_{0}), 1-b_{1|0}(1|w_{0}); \rho\}] \mathrm{d} w_{0}\\ &=& 2-{\int}_{0}^{\infty}C_{N}\{b_{1|0}(1|w_{0}), b_{1|0}(1|w_{0}); \rho\} \mathrm{d} w_{0}\\ &=& 2 - {\int}_{0}^{\infty}b_{1|0}(1|w_{0}) \mathrm{d} w_{0} + \left( \frac{1-\rho}{\pi}\right)^{1/2}{\int}_{0}^{\infty} \phi[{\Phi}^{-1}\{b_{1|0}(1|w_{0})\}] \mathrm{d} w_{0} + O((1-\rho)^{3/2})\\ &=& 1 + \left( \frac{1-\rho}{\pi}\right)^{1/2}{\int}_{0}^{\infty} \phi[{\Phi}^{-1}\{b_{1|0}(1|w_{0})\}] \mathrm{d} w_{0} + O((1-\rho)^{3/2}). \end{array} $$

□

1.5 A.5 Computation of V _j,k(w _j,w _k)

Let I_j,k(w₀) = 1 − C_N{1 − b_j|0(w_j|w₀), 1 − b_k|0(w_k|w₀); ρ_j,k}. We assume that, as $w_{0} \to \infty$, $b_{j|0}(w_{j}|w_{0}) = \ell _{j}(w_{0}) w_{0}^{-\phi _{j}} + o(w_{0}^{-\phi _{j}})$ and $b_{k|0}(w_{k}|w_{0}) = \ell _{k}(w_{0}) w_{0}^{-\phi _{k}} + o(w_{0}^{-\phi _{k}})$, ϕ_j,ϕ_k > 1, where ℓ_j and ℓ_k are slowly varying functions. It follows that $I_{j,k}(w_{0}) = \ell _{j,k}(w_{0})w_{0}^{-\phi _{j,k}}$ as $w_{0} \to \infty$ where $\phi _{j,k} = \min \limits (\phi _{j}, \phi _{k})$ and ℓ_j,k is a slowly varying function. The integrand I_j,k(w₀) has a slow rate of decay in the tail and standard numerical integration methods used to compute $V_{j,k}(w_{j},w_{k}) = {\int \limits }_{0}^{\infty } I_{j,k}(w_{0}) \mathrm {d} w_{0}$ may not be efficient.

To make computations more efficient, we can write

$$ V_{j,k}(w_{j},w_{k}) = {{\int}_{0}^{1}} I_{j,k}(w_{0}) \mathrm{d} w_{0} + {\int}_{1}^{\infty} I_{j,k}(w_{0}) \mathrm{d} w_{0} = {{\int}_{0}^{1}} I_{j,k}(w_{0}) \mathrm{d} w_{0} + {\alpha{\int}_{0}^{1}} I_{j,k}(z_{0}^{-\alpha}) z_{0}^{-\alpha - 1}\mathrm{d} z_{0}, $$

where the first integral has finite integration limits and bounded integrand. The second integrand can take large values if z₀ is close to zero. We therefore need to select the smallest α > 0 such that $I_{j,k}^{*}(z_{0}) = I_{j,k}(z_{0}^{-\alpha })z_{0}^{-\alpha - 1} < \infty$ for 0 ≤ z₀ ≤ 1. We have $I^{*}_{j,k}(z_{0}) = \ell _{j,k}(z_{0}^{-\alpha })z_{0}^{(\phi _{j,k}-1)\alpha -1}$ as z₀ → 0 and one can select $\alpha = \alpha _{j,k} = \{\phi _{j,k}-1\}^{-1}$. Now Gauss-Legendre quadrature (Stroud and Secrest 1966) can be used to evaluate the two integrals and compute V_j,k(w_j,w_k).

The assumption about the tail behavior of b_j|0 holds for many copulas with the upper tail dependence. For the reflected Clayton copula with parameter 𝜃_j,

$$ \begin{array}{@{}rcl@{}} b_{j|0}(w_{j}|w_{0}) &=& \left\{1+(w_{0}/w_{j})^{\theta_{j}}\right\}^{-1-1/\theta_{j}} \\&=& (w_{0}/w_{j})^{-1-\theta_{j}} + o(w_{0}^{-1-\theta_{j}}), \quad \text{as } w_{0} \to \infty, \end{array} $$

and therefore $\alpha _{j,k} = 1/\min \limits (\theta _{j}, \theta _{k})$. For the Gumbel copula with parameter 𝜃_j,

$$ b_{j|0}(w_{j}|w_{0}) = 1-\left\{1+(w_{j}/w_{0})^{\theta_{j}}\right\}^{-1+1/\theta_{j}} = (1-1/\theta_{j})(w_{0}/w_{j})^{-\theta_{j}} + o(w_{0}^{-\theta_{j}}), \quad \text{as } w_{0} \to \infty, $$

and therefore $\alpha _{j,k} = 1/\{\min \limits (\theta _{j}, \theta _{k})-1\}$.

Similar ideas can be used to compute the derivatives of the stable tail dependence function. We found that the same transformation works very well in this case and that this change of variables greatly improves the accuracy of numerical integration and n_q = 35 quadrature points is sufficient to compute the density $c_{j,k}^{\text {EV}}$ in most cases.

1.6 A.6 Parameter estimation for $C_{\mathbf {U}}^{\text {EV}}$ with Clayton linking copulas

Here we provide more details for $C_{\mathbf {U}}^{\text {EV}}$ in Section 3.2 with Clayton linking copulas. Note that V_j,k(1, 1) = 1 if 𝜃_j = 𝜃_k. Without loss of generality, we now assume that 𝜃_j > 𝜃_k for the (j,k) margin. We have c(1,v; 𝜃) = (𝜃 + 1)v^𝜃 and

$$ V_{j,k}(1,1) = {\int}_{0}^{v_{j,k}} c(1,v_{0};\theta_{k})\mathrm{d} v_{0} + {\int}_{v_{j,k}}^{1} c(1,v_{0};\theta_{j})\mathrm{d} v_{0} = 1 - C(v_{j,k}|1; \theta_{j}) + C(v_{j,k}|1; \theta_{k}), $$

where the conditional Clayton copula C(v|1; 𝜃) = v^𝜃+ 1 and v_j,k ∈ (0, 1) satisfies

$$ c(1,v_{j,k}; \theta_{j}) = c(1,v_{j,k}; \theta_{k}) \quad \Rightarrow \quad v_{j,k} = \left( \frac{\theta_{k}+1}{\theta_{j}+1}\right)^{\frac{1}{\theta_{j} - \theta_{k}}} , $$

and therefore

$$ V_{j,k}(1,1) = 1 + \frac{(\vartheta-1)^{\vartheta-1}}{\vartheta^{\vartheta}}, \quad \vartheta = \frac{\theta_{j}+1}{\theta_{j} - \theta_{k}} . $$

Similarly, one can show that the copula $C_{\mathbf {U}}^{\text {EV}}$ and its lower-dimensional marginals depend on 𝜗(𝜃_j,𝜃_k) = (𝜃_j + 1)/(𝜃_j − 𝜃_k), or, equivalently, on 𝜗^∗(𝜃_j,𝜃_k) = {𝜗(𝜃_j,𝜃_k) − 1}/𝜗(𝜃_j,𝜃_k) = (𝜃_k + 1)/(𝜃_j + 1) for 1 ≤ k < j ≤ d. The model is therefore non-identifiable and one can fix one parameter and estimate the remaining d − 1 parameters.

If the order of variables is ignored, the model is still non-identifiable even if one parameter is fixed.

Example 5

Assume that d = 3 and 𝜃 = (0.5, 1, 2)^⊤. We generate a sample of size N = 100 from $C_{\mathbf {U}}^{\text {EV}}$ assuming Σ is a matrix of ones. We fix 𝜃₂ = 1 and find that the objective function (4) attains its minimum at $\widehat {\boldsymbol {\theta }} = (0.456, 1, 2.584)^{\top }$ and $\tilde {\boldsymbol {\theta }} = (1.747, 1, 0.116)^{\top }$. We can see that

$$ \frac{\widehat{\boldsymbol{\theta}}_{1}+1}{\widehat{\boldsymbol{\theta}}_{2}+1} = \frac{\tilde{\boldsymbol{\theta}}_{2}+1}{\tilde{\boldsymbol{\theta}}_{1}+1} , \quad \frac{\widehat{\boldsymbol{\theta}}_{1}+1}{\widehat{\boldsymbol{\theta}}_{3}+1} = \frac{\tilde{\boldsymbol{\theta}}_{3}+1}{\tilde{\boldsymbol{\theta}}_{1}+1} , \quad \frac{\widehat{\boldsymbol{\theta}}_{2}+1}{\widehat{\boldsymbol{\theta}}_{3}+1} = \frac{\tilde{\boldsymbol{\theta}}_{3}+1}{\tilde{\boldsymbol{\theta}}_{2}+1} . $$

To select the right solution, one can check bivariate scatter plots: if 𝜃_j > 𝜃_k for the (j,k) margin, the marginal density is zero around the corner (0,1) and the density is skewed to the lower right corner. Figure 5 shows scatter plots for the simulated data set. The plots indicate that 𝜃₁ < 𝜃₂ < 𝜃₃ and therefore $\widehat {\boldsymbol {\theta }}$ should be selected.

Fig. 5

Bivariate scatter plots for the data set simulated from $C_{\mathbf {U}}^{\text {EV}}$ with degenerate Σ (matrix of ones) and Clayton linking copulas with 𝜃 = (0.5,1,2)^⊤