Application to three-dimensional vine copulas

G⁽ⁿ⁾₁ (m₁), . . . , G⁽ⁿ⁾_d (m_d)

=C(Φ(m₁), . . . ,Φ(m_d))ⁿ. For the corresponding joint density g⁽ⁿ⁾ of the block maxima over Z_i we also obtain an explicit expression as a corollary of Theorem 4.1, where once again Sklar’s Theorem is applied.

Corollary 4.3. Form_j ∈R, j = 1, . . . , d, we have

g⁽ⁿ⁾(m₁, . . . , m_d) =

d

Y

j=1

ϕ(m_j)

!

·

d∧n

X

j=1

( n!

(n−j)!·C(Φ(m₁), . . . ,Φ(m_d))^n−j

× X

P∈Sd,j

Y

I∈P

∂_IC(Φ(m₁), . . . ,Φ(m_d)) )

. (4.7)

4.3. Application to three-dimensional vine copulas

As an application of the results in the previous section we will consider vine copulas in three dimensions and show why they are a particularly useful class of model for this purpose. We have seen in Chapter 2 that vine copulas allow for product expressions of the density. Since in three dimensions all (three) structures are equivalent in the sense that they can be obtained by relabeling the variables, we only consider the following decomposition for the remained of the chapter:

c(u₁, u₂, u₃) = c_1,2(u₁, u₂)c_2,3(u₂, u₃)c_1,3;2(C_1|2(u₁|u₂), C_3|2(u₃|u₂);u₂). (4.8) We have seen that we need the copula functionC and its partial derivatives in order to calculate the copula density of the block maxima using in Theorem 4.1. These expressions are derived in Proposition 4.4. The corresponding proof can be found in Appendix B.2.

Proposition 4.4. For a three-dimensional vine copula with decomposition as in Equa-tion 4.8 the partial derivatives of the copula funcEqua-tion can be obtained as follows:

1. C(u₁, u₂, u₃) =Ru2

0 C_1,3;2 C_1|2(u₁|v₂), C_3|2(u₃|v₂);v₂ dv₂; 2. a) ∂₁C(u₁, u₂, u₃) = Ru2

0 C_3|1;2(C_1|2(u₁|v₂), C_3|2(u₃|v₂);v₂)c_1,2(u₁, v₂) dv₂; b) ∂₂C(u₁, u₂, u₃) = C_1,3;2(C_1|2(u₁|u₂), C_3|2(u₃|u₂);u₂);

c) ∂₃C(u₁, u₂, u₃) = Ru2

0 C_1|3;2(C_1|2(u₁|v₂), C_3|2(u₃|v₂);v₂)c_2,3(v₂, u₃) dv₂; 3. a) ∂₁₂C(u₁, u₂, u₃) =C_3|1;2(C_1|2(u₁|u₂), C_3|2(u₃|u₂);u₂)c_1,2(u₁, u₂);

b) ∂₁₃C(u₁, u₂, u₃) =Ru2

0 c_1,3;2(C_1|2(u₁|v₂), C_3|2(u₃|v₂);v₂)c_2,3(v₂, u₃)c_1,2(u₁, v₂) dv₂; c) ∂₂₃C(u₁, u₂, u₃) =C_1|3;2(C_1|2(u₁|u₂), C_3|2(u₃|u₂);u₂)c_2,3(u₂, u₃);

4. c(u₁, u₂, u₃) = c_1,2(u₁, u₂)c_2,3(u₂, u₃)c_1,3;2(C_1|2(u₁|u₂), C_3|2(u₃|u₂);u₂).

Proposition 4.4 shows that the copula density corresponding to the three-dimensional vector of block maxima based on an arbitrary vine copula is numerically tractable since only one-dimensional integration is required. In particular this allows a numerical treat-ment for the block size n in a finite setting.

In the following three examples we will work with simplified vine copulas. In order to illustrate the influence of the block size on the dependence structure of the block maxima we consider the corresponding copula density c_M(n) for block sizes n = 1,10,50,1000.

Note that for a block size of n = 1 the copula density of the block maxima simply is the copula density c of the underlying distribution, i.e. c_M(1) = c, since taking maxima over blocks of size one does not have any impact. To visualize the copula density of the block maxima c_M(n) for different block sizes n we consider the contour surfaces of these three-dimensional copula densities. This approach is the natural extension of considering contour lines for bivariate copulas. Since copula densities on the original [0,1]-scale are difficult to interpret and comparisons between different models would hardly be possible one usually uses univariate standard normal margins when considering contour plots. Therefore, we also use univariate standard normal margins for our three-dimensional contour surface plots. The contour levels are fixed to be 0.015, 0.035, 0.075 and 0.110 for each plot (from outer to inner surface) in order to enable comparisons between different block sizes and examples. We show the contour surfaces from three different angles such that we get a representative impression of the shape of the contours. This way of illustrating three-dimensional copula densities is inspired by Killiches et al. (2017a). Furthermore, we apply our approach to a three-dimensional hydrological data set.

Example 4.5.The first example we present is a three-dimensional Gaussian vine, i.e. all three pair-copulas are bivariate Gaussian copulas. As parameters we choose ρ_1,2 = 0.71, ρ_2,3 = 0.78 and ρ_1,3;2 = 0.52 corresponding to Kendall’s τ values of τ_1,2 = 0.50 and τ_2,3 = 0.57, τ_1,3;2 = 0.35.

Figure 4.1 shows the copula density of the block maxima of this vine copula with standard normal margins for block sizes n = 1,10,50,1000. Each row represents one block size and contains three plots from different angles. We see that for increasing block size the contours tend to concentric spheres around the origin. This is what the contour surfaces of the independence copula (with standard normal margins) look like in three dimensions. This is also what one would expect: Recall from Section 2.2.3 that every Gaussian vine is also a Gaussian copula. Sibuya (1960) proved that any Gaussian copula

Figure 4.1.: Plot of the contour surfaces of the copula density of block maxima of a three-dimensional Gaussian vine with standard normal margins (τ1,2 = 0.50,τ2,3 = 0.57, τ_1,3;2 = 0.35) from different angles (columns). The rows correspond to block sizes

lies in the domain of attraction of the independence copula, i.e. the block maxima of any Gaussian copula converge to the independence copula. Thus, our empirical findings are in line with the theory. H¨usler and Reiss (1989) showed that in order to achieve that the distribution of the block maxima of a multivariate Gaussian distribution converges to a non-trivial limiting distribution, a proper scaling of the margins and the correlation coefficients is necessary. This will be discussed in Section 4.4.

Example 4.6. As a second example we take a three-dimensional Clayton vine, i.e. all three pair-copulas are now bivariate Clayton copulas. As parameters we choose δ_1,2 = 6, δ_2,3 = 7.09 and δ_1,3;2 = 4.67 corresponding to Kendall’s τ values of τ_1,2 = 0.75 and τ_2,3 = 0.78, τ_1,3;2 = 0.70.

Similar to the Gaussian example Figure 4.2 shows the copula density of the block max-ima of this vine copula with standard normal margins for block sizes n = 1,10,50,1000.

Each row represents one block size and contains three plots from different angles. We see that for increasing block size the contours tend to concentric spheres around the ori-gin corresponding to the contour surfaces of the independence copula, which are already (approximately) reached for n= 1000.

Remark 4.7. Even though it is not known whether all Clayton vines lie in the domain of attraction of the independence copula, one can show that the Clayton copula, which can be represented as a Clayton vine with specific parameter restrictions (cf. Section 2.2.3), lies in the domain of attraction of the independence copula. According to Gudendorf and Segers (2010) an Archimedean copula with generator ϕ lies in the domain of attraction of the Gumbel copula with parameter

θ :=−lim

s↓0

sϕ⁰(1−s)

ϕ(1−s) ∈[1,∞)

if the limit exists. For the Clayton copula this limit is equal to 1. Therefore, the copula of the block maxima of a Clayton copula converges to the Gumbel copula with θ= 1, which corresponds to the independence copula.

Example 4.8. Having investigated two examples where all three pair-copulas belonged to the same family we will now consider a mixed vine copula with very high strengths of dependence specified as follows: c_1,2 is a Frank copula with parameter 8 (τ_1,2 = 0.8), c_2,3 is a Clayton copula with parameter 18.19 (τ_2,3 = 0.8) and c_1,3;2 is a Gaussian copula with parameter 0.95 (τ_1,3;2 = 0.8). As before we show the copula density of the block maxima of this vine copula with standard normally distributed margins for block sizes n = 1,10,50,1000 in Figure 4.3. Each row represents one block size and contains three plots from different angles. We see that due to the high parameter values there can still be detected some non-negligible dependence for block sizes n = 50 and even n = 1000.

Figure 4.2.: Plot of the contour surfaces of the copula density of block maxima of a three-dimensional Clayton vine with standard normal margins (τ1,2 = 0.75,τ2,3 = 0.78, τ_1,3;2 = 0.70) from different angles (columns). The rows correspond to block sizes

Figure 4.3.: Plot of the contour surfaces of the copula density of block maxima of a three-dimensional Frank-Clayton-Gaussian vine with standard normal margins (τ1,2 = 0.80, τ_2,3 = 0.80,τ_1,3;2 = 0.80) from different angles (columns). The rows correspond to

Therefore, we further increased block sizes ton = 10⁶andn = 10⁸ in Figure 4.4. Although there is still some dependence left, as before we can see the clear tendency to the contours of the independence copula.

Figure 4.4.: Plot of the contour surfaces of the copula density of block maxima of a three-dimensional Frank-Clayton-Gaussian vine with standard normal margins (τ1,2 = 0.80, τ_2,3 = 0.80,τ_1,3;2 = 0.80) from different angles (columns). The rows correspond to block sizesn= 10⁶,10⁸.

Example 4.9. Hydrology is one of the areas where block maxima are important. Espe-cially, the water levels of rivers can be interesting when it comes to analyzing the risk of floods. We consider a three-dimensional data set containing the water levels of rivers in and around Munich, Germany, from August 1, 2007 to July 31, 2013. The data has been taken from Bavarian Hydrological Service (http://www.gkd.bayern.de). The three variables denote the differences of the 12 hour average water levels at the following three measuring points: the Isar measured in Munich, the Isar measured in Baierbrunn (south of Munich) and the Schwabinger Bach measured in Munich (a small stream entering the Isar in Garching, north of Munich). Since we only consider the hydrological winter (November 1 to April 30), we have 2176 data points.

First, we transform the margins to the unit interval applying the probability integral transform with the empirical marginal distribution functions. Then, we fit the dependence

Figure 4.5.: Plot of the contour surfaces of the copula density for block maxima of the water level differences for one day (first row), one week (second row), one month (third row) and one winter (fourth row) with standard normal margins.

structure with a vine copula usingRVineStructureSelect⁵:c_1,2is estimated to be a Frank copula with a Kendall’s τ of τ_1,2 = 0.76, c_2,3 is a Frank copula with τ_2,3 = 0.23 and c_1,3;2 is a Gaussian copula with τ_1,3;2 = −0.18. Now we are interested in the resulting copula density of the maxima for one day (n = 2), one week (n = 14), one month (n = 60) and one winter (n = 362). The corresponding contour surfaces are plotted in Figure 4.5.

Similar to the examples from above we see that with increasingnthe observed dependence structure tends to the independence copula. In case of the considered rivers this means that the maximal differences of the 12 hour average water levels over the entire winter are (almost) independent.