Assessing and Modeling Asymmetry in Bivariate Continuous Data

Abstract

A bivariate copula is the cumulative distribution function of a pair (U, V ) of uniform random variables. This copula is said to be symmetric if and only if (V, U) and (U, V ) have the same distribution. Many standard bivariate parametric families of copulas have this property; Archimedean and meta-elliptical copulas are prime examples. In practice, however, dependence is often asymmetric. This paper revisits key aspects of this issue from a modeling perspective. Measures of asymmetry and rank-based estimators thereof are discussed, along with recently proposed tests of symmetry. Several techniques for the construction of asymmetric dependence structures are critically reviewed. A hydrological data set is used for illustration purposes.

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5.1 Introduction

Let (X, Y ) be a pair of continuous random variables and let its marginal and joint cumulative distribution functions be denoted, at each $x,y \in \mathbb{R}$, by F(x) = Pr(X ≤ x), G(y) = Pr(Y ≤ y), and H(x, y) = Pr(X ≤ x, Y ≤ y), respectively. Sklar’s Representation Theorem states that there exists a unique copula C such that, for all $x,y \in \mathbb{R}$,

$$\displaystyle{ H(x,y) = C\{F(x),G(y)\}. }$$

(5.1)

In fact, C is the cumulative distribution function of the pair (U, V ) = (F(X), G(Y )) having uniform margins on the interval (0, 1).

In applications, H is typically unknown. A convenient way to model it consists of expressing H in the form (5.1) and assuming that F, G, and C belong to specific parametric classes of distributions. For instance, one might take F Gaussian, G Pareto, and C from the Farlie–Gumbel–Morgenstern family of copulas.

Many parametric classes of bivariate copulas have been proposed; see, e.g., [26, 36] for extensive lists. Most classical copulas are symmetric, however. This property, also referred to as exchangeability, means that for all u, v ∈ [0, 1],

$$\displaystyle{ C(u,v) = C(v,u). }$$

(5.2)

When this condition fails for some u, v ∈ [0, 1], C is said to be asymmetric, or non-exchangeable. Note that in the copula literature, the term “asymmetric” sometimes refers to the lack of radial symmetry, and particularly the presence of right-tail but no left-tail dependence; see, e.g., [39, 47] for applications in asset allocation and equity markets. These types of asymmetry are not further considered here.

Asymmetric dependence, i.e., failure of (5.2) for some u, v ∈ [0, 1], typically occurs when there is a causality relationship between the variables. Consider, for instance, the monthly average flow rate (in m³ ∕ s) of the Inn and Danube rivers [48], measured at the Schärding and Nagymaros stations, respectively. The stylized map in Fig. 5.1 suggests that high discharge X of the Inn (upstream) is likely to imply high discharge Y of the Danube (downstream), but not necessarily vice versa. Asymmetry in the dependence between X and Y is apparent from Fig. 5.2, which shows approximate scatter plots of the copulas of the raw (left) and de-trended data (right).

Fig. 5.1

Stylized map showing the location of the Schärding and Nagymaros measurement stations on the Inn and Danube rivers, respectively

Fig. 5.2

Rank plots of pairs of raw data (left panel) and de-trended (right panel) monthly average flow rate at the Schärding and Nagymaros stations in the period 1936–1991

Fig. 5.3

Plot of the difference $\tau _{\max }(\lambda ,\kappa ) -\tau _{\min }(\lambda ,\kappa )$ as a function of λ and κ

The purpose of this paper is to provide a critical review of the literature on asymmetric copulas and to contribute to it in modest ways. Its structure reflects the progression of a statistical analysis of asymmetric dependence; the river discharge data are used throughout for illustration. Measures of asymmetry and their estimates are discussed in Sect. 5.2. Tests of asymmetry based on these measures are presented in Sect. 5.3. Techniques for constructing asymmetric copulas are reviewed in Sect. 5.4. Concluding comments are given in Sect. 5.5.

5.2 Measures of Asymmetry

Given p ≥ 1, a natural measure of asymmetry in a copula C is given by

$$\displaystyle{\mu _{p}(C) ={ \left \{\int _{0}^{1}\int _{ 0}^{1}\vert C(u,v) - C(v,u){\vert }^{p}\mbox{ d}v\mbox{ d}u\right \}}^{1/p},}$$

i.e., the L _p distance between C and its “transpose” ${C}^{\top }$ defined, for all u, v ∈ [0, 1], by ${C}^{\top }(u,v) = C(v,u)$. One could also consider letting p → ∞, which leads to

$$\displaystyle{\mu _{\infty }(C) =\sup _{(u,v)\in {[0,1]}^{2}}\vert C(u,v) - C(v,u)\vert.}$$

These measures were first discussed in a copula context by Klement and Mesiar [28] and Nelsen [37]. Durante et al. [13] show that the measures μ _p , with p ∈ [1, ∞], satisfy a series of natural axioms for measures of non-exchangeability. To be specific, let $\mathcal{C}$ be the class of bivariate copulas and, for any $C \in \mathcal{C}$ and u, v ∈ [0, 1], set

$$\displaystyle{ \bar{C}(u,v) = u + v - 1 + C(1 - u,1 - v). }$$

Durante et al. [13] then show that:

(B1):

There exists K > 0 such that 0 ≤ μ _p (C) ≤ K for all $C \in \mathcal{C}$.

(B2):

μ _p (C) = 0 if and only if C = C ^⊤.

(B3):

$\mu _{p}(C) =\mu _{p}({C}^{\top })$ for all $C \in \mathcal{C}$.

(B4):

$\mu _{p}(C) =\mu _{p}(\bar{C})$ for all $C \in \mathcal{C}$.

(B5):

If $C_{n} \in \mathcal{C}$ and if $C_{n} \rightarrow C \in \mathcal{C}$ uniformly as n → ∞, then $\mu _{p}(C_{n}) \rightarrow \mu _{p}(C)$.

Siburg and Stoimenov [43] consider alternative measures of the form

$$\displaystyle{\delta (C) = \frac{\|C{_{s}\|}^{2} -\| C{_{a}\|}^{2}} {\|{C\|}^{2}} ,}$$

which take into account both the symmetric part $C_{s} = (C + {C}^{\top })/2$ and the asymmetric part $C_{a} = (C - {C}^{\top })/2$ of C. The norm $\|\cdot \|$ can be arbitrary, but the authors focus on a modified Sobolev norm. Given that their measure is equivalent to μ _∞ for the uniform norm and that the Sobolev norm seems to be of limited statistical use, their idea is not considered further here.

5.2.1 Maximal Asymmetry

It is clear that whatever p ∈ [1, ∞], μ _p (C) = 0 if and only if C is symmetric. Because | C(u, v) | ≤ 1 for any u, v ∈ [0, 1], it is immediate that μ _p (C) ≤ 2 for any choice of C and p ∈ [1, ∞]. This bound is, however, never attained and as such unsuitable for standardization purposes. To obtain sharper bounds, Klement and Mesiar [28] and Nelsen [37] prove that for any C and u, v ∈ [0, 1], | C(u, v) − C(v, u) | ≤ Δ(u, v) where

$$\displaystyle{\Delta (u,v) =\min (u,v,1 - u,1 - v,\vert v - u\vert ).}$$

This implies that for any C,

$$\displaystyle{\mu _{\infty }(C) \leq \kappa _{\infty } =\sup _{u,v\in [0,1]}\vert \Delta (u,v)\vert = \frac{1} {3}\,,}$$

while for any p ∈ [1, ∞),

$$\displaystyle{\mu _{p}(C) \leq \kappa _{p} ={ \left \{\int _{0}^{1}\int _{ 0}^{1}\Delta {(u,v)}^{p}\mbox{ d}v\mbox{ d}u\right \}}^{1/p} ={ \left \{ \frac{2 \times {3}^{-p}} {(p + 1)(p + 2)}\right \}}^{1/p}}$$

as computed by Durante et al. [13]. Although Proposition 3.1 in [28] states that no copula exists such that $\vert C(u,v) - C(v,u)\vert = \Delta (u,v)$ for all u, v ∈ [0, 1], the bound $\kappa _{\infty }$ turns out to be sharp. The following result is excerpted from [37].

Proposition 5.1.

If C is an arbitrary copula, then $\mu _{\infty }(C) = 1/3$ if and only if either (i) $C(\frac{1} {3}, \frac{2} {3}) = \frac{1} {3}$ and $C(\frac{2} {3}, \frac{1} {3}) = 0$ , or (ii) $C(\frac{1} {3}, \frac{2} {3}) = 0$ and $C(\frac{2} {3}, \frac{1} {3}) = \frac{1} {3}$ .

By Proposition 5.1, $\mu _{\infty }(C) = 1/3$ whenever C or ${C}^{\top }$ places one third of its probability mass in each of the rectangles $[0,1/3] \times [1/3,2/3]$, $[1/3,2/3] \times [2/3,1]$ and $[2/3,1] \times [0,1/3]$. There are infinitely many such copulas, one prime example being

$$\displaystyle{C_{\infty }(u,v) =\max \{\max (u + v - 1,0),\min (u,v - 1/3)\},}$$

which is a singular copula whose support consists of two line segments, one from (0, 1 ∕ 3) to (2 ∕ 3, 1) and the other from $(2/3,1/3)$ to (1, 0).

When p ∈ [1, ∞), the matter is less clear, however. Although Proposition 4 of Durante et al. [13] guarantees that there exists a copula C _p for which μ _p is maximized, it is not known what C _p looks like and whether $\mu _{p}(C_{p}) =\kappa _{p}$.

Finally, there are several interesting relationships between symmetry and association. As explained by Nelsen [37], Proposition 5.1 leads to bounds on the Spearman and Kendall measures of association given, respectively, by

$$\displaystyle{\rho (C) = -3 + 12\int _{0}^{1}\int _{ 0}^{1}C(u,v)\mathrm{d}v\mathrm{d}u,\quad \tau (C) = -1 + 4\int _{ 0}^{1}\int _{ 0}^{1}C(u,v)\mathrm{d}C(u,v).}$$

Nelsen concludes that whenever $\mu _{\infty }(C) = 1/3$, $\rho (C) \in [-5/9,-1/3]$ while $\tau (C) \in [-5/9,1/9]$. This supports the statement of De Baets et al. [6] who write that “positive dependence plays in favor of symmetry.” These authors find that when C is positive quadrant dependent, i.e., if for all u, v ∈ [0, 1], C(u, v) ≥ u v, the measure of asymmetry μ _∞ (C) is at most $3 - 2\sqrt{2} \approx 0.172$. They also identify the copula for which the bound is achieved. When C has the stronger positive dependence property called stochastic increasingness in both arguments, Durante and Papini [10] show that the upper bound for μ _∞ is given by $(5\sqrt{5} - 11)/2 \approx 0.09$.

When C is negative quadrant dependent, i.e., if for all u, v ∈ [0, 1], C(u, v) ≤ u v, Durante and Papini [11] show that $\mu _{\infty }(C) \leq \sqrt{5} - 2 \approx 0.236$. They also find that when C is stochastically decreasing in both arguments, $\mu _{\infty }(C) \leq 3 - 2\sqrt{2} \approx 0.172$ and conclude that symmetric concepts of strong negative association decrease the level of possible asymmetry.

5.2.2 Estimates

Given a random sample $(X_{1},Y _{1}),\ldots ,(X_{n},Y _{n})$ from an unknown distribution H with unique underlying copula C, the question arises as to how to estimate μ _p (C). To this end, let the (rescaled) empirical distribution of H be defined, for all $x,y \in \mathbb{R}$, by

$$\displaystyle{H_{n}(x,y) = \frac{1} {n + 1}\sum _{i=1}^{n}\mathbf{1}(X_{ i} \leq x,Y _{i} \leq y),}$$

and let F _n and G _n denote its margins. The more traditional definition of H _n involving division by n rather than n + 1 leads to slightly different expressions in finite samples, but this has no effect asymptotically.

For each i ∈ { 1, …, n}, the pair $(\hat{U}_{i},\hat{V }_{i}) = (F_{n}(X_{i}),G_{n}(Y _{i}))$ may be viewed as a pseudo-observation from C. A natural estimate of C is then given by the empirical copula $\hat{C}_{n}$ defined, at every u, v ∈ [0, 1], by

$$\displaystyle{\hat{C}_{n}(u,v) = \frac{1} {n}\sum _{i=1}^{n}\mathbf{1}(\hat{U}_{ i} \leq u,\hat{V }_{i} \leq v).}$$

Note that $\hat{C}_{n}$ is rank-based because for all i ∈ { 1, …, n}, $(n + 1)\hat{U}_{i}$ is the rank of X _i among $X_{1},\ldots ,X_{n}$ and similarly, $(n + 1)\hat{V }_{i}$ is the rank of Y _i among $Y _{1},\ldots ,Y _{n}$.

Plug-in estimates of μ _∞ (C) and μ _p (C) for p ∈ [1, ∞) are then given by

$$\displaystyle\begin{array}{rcl} \mu _{\infty }(\hat{C}_{n})& =& \sup _{(u,v)\in {[0,1]}^{2}}\left \vert \hat{C}_{n}(u,v) -\hat{ C}_{n}(v,u)\right \vert , {}\\ \mu _{p}(\hat{C}_{n})& =&{ \left \{\int _{0}^{1}\int _{ 0}^{1}{\left \vert \hat{C}_{ n}(u,v) -\hat{ C}_{n}(v,u)\right \vert }^{p}\mbox{ d}v\,\mbox{ d}u\right \}}^{1/p}. {}\\ \end{array}$$

Their consistency stems from the fact that the empirical copula $\hat{C}_{n}$ is itself a consistent estimator of C, provided that C is regular in the following sense [41].

Definition 5.1.

A bivariate copula C is said to be regular if

• (i) the partial derivatives $\dot{C}_{1}(u,v) = \partial C(u,v)/\partial u$ and $\dot{C}_{2}(u,v) = \partial C(u,v)/\partial v$ exist everywhere on [0, 1]², where by convention, one-sided derivatives are used at the boundary points;

• (ii) $\dot{C}_{1}$ is continuous on (0, 1) ×[0, 1] and $\dot{C}_{2}$ is continuous on [0, 1] ×(0, 1).

As illustrated by Segers [41], all copulas commonly used in practice satisfy this condition. Assuming henceforth that C is regular and observing that μ _p is a continuous functional of C for each p ∈ [1, ∞], the Continuous Mapping Theorem readily implies the following result.

Proposition 5.2.

If C is a regular copula, then for all p ∈ [1,∞], $\mu _{p}(\hat{C}_{n})$ converges in probability to μ _p (C) as n →∞.

The estimator $\mu _{\infty }(\hat{C}_{n})$ is easier to compute than it appears because

$$\displaystyle{\mu _{\infty }(\hat{C}_{n}) =\max _{i,j\in \{1,\ldots ,n\}}\left \vert \hat{C}_{n}\left ( \frac{i} {n + 1}, \frac{j} {n + 1}\right ) -\hat{ C}_{n}\left ( \frac{j} {n + 1}, \frac{i} {n + 1}\right )\right \vert.}$$

As shown by Genest et al. [23], one has also

$$\displaystyle{\mu _{2}^{2}(\hat{C}_{ n}) = \frac{2} {{n}^{2}}\sum _{i=1}^{n}\sum _{ j=1}^{n}(1-\hat{U}_{ i}\vee \hat{U}_{j})(1-\hat{V }_{i}\vee \hat{V }_{j})-(1-\hat{U}_{i}\vee \hat{V }_{j})(1-\hat{V }_{i}\vee \hat{U}_{j}),}$$

where for arbitrary $a,b \in \mathbb{R}$, a ∨ b = max(a, b). For other values of p, $\mu _{p}(\hat{C}_{n})$ may be tedious to compute. To circumvent this problem, one might consider an alternative empirical measure of asymmetry given by

$$\displaystyle{\nu _{p}(\hat{C}_{n}) ={ \left \{\int _{0}^{1}\int _{ 0}^{1}{\left \vert \hat{C}_{ n}(u,v) -\hat{ C}_{n}(v,u)\right \vert }^{p}\mbox{ d}\hat{C}_{ n}(u,v)\right \}}^{1/p}.}$$

Because $\hat{C}_{n}$ is the joint distribution function of a counting probability measure,

$$\displaystyle{\nu _{p}^{p}(\hat{C}_{ n}) = \frac{1} {n}\sum _{i=1}^{n}{\left \vert \hat{C}_{ n}(\hat{U}_{i},\hat{V }_{i}) -\hat{ C}_{n}(\hat{V }_{i},\hat{U}_{i})\right \vert }^{p}.}$$

Furthermore, the following result is showed in the Appendix.

Proposition 5.3.

For p ∈ [1,∞), $\nu _{p}(\hat{C}_{n})$ converges in probability, as n →∞, to

$$\displaystyle{ \nu _{p}(C) ={ \left \{\int _{0}^{1}\int _{ 0}^{1}{\left \vert C(u,v) - C(v,u)\right \vert }^{p}\mathrm{d}C(u,v)\right \}}^{1/p}. }$$

(5.3)

Equation (5.3) defines a new population measure of asymmetry in the sense of Durante et al. [13]. This result, proved in the Appendix, is formally recorded below.

Proposition 5.4.

For any p ∈ [1,∞), ν _p satisfies the axioms (B1)–(B5).

While ν _p is easier to estimate than μ _p , a realistic (let alone sharp) upper bound for its value is unknown at present. Such a bound would be useful to compare values of ν _p across copulas and with other standardized measures of asymmetry.

5.2.3 Illustration

Consider the monthly average flow rate (in m³ ∕ s) of the Inn and Danube rivers, as observed at the Schärding and Nagymaros measurement stations in the 55-year period extending from 1936 to 1991. The 660 pairs of normalized ranks $(\hat{U}_{i},\hat{V }_{i})$ for the raw data are displayed in Fig. 5.2 (left). Though the plot suggests asymmetry, the estimates of μ _∞ , μ ₂ and ν ₂ are not particularly large:

$$\displaystyle{3\,\mu _{\infty }(\hat{C}_{n}) \approx 0.127,\quad \sqrt{54}\,\mu _{2}(\hat{C}_{n}) \approx 0.083,\quad \nu _{2}(\hat{C}_{n}) \approx 0.014.}$$

For comparison purposes, μ _∞ and μ ₂ were divided by $\kappa _{\infty } = 1/3$ and $\kappa _{2} = 1/\sqrt{54}$, respectively. This means that 3 μ _∞ takes values in the entire interval [0, 1], while it is only known that $\sqrt{54}\,\mu _{2} \leq 1$. Although $\kappa _{2}$ may not be sharp, it seems to give a reasonable standardization in practice. As an upper bound for ν ₂ is unknown, no standardization was made. Given that the values of μ ₂ and ν ₂ are often close, κ ₂ can be used as a rule of thumb adjustment; this yields $\sqrt{ 54}\,\nu _{2}(\hat{C}_{n}) \approx 0.0997$.

Hydrological data typically exhibit time trends. Such is the case here and hence the relationship between the two variables may be confounded with serial dependence. To eliminate this effect, Bacigál et al. [2] suggest de-trending the series by AR(1) models. The resulting residuals exhibit no further significant autocorrelation and can thus be used to study the time-invariant dependence between the variables (assuming there is one). The rank plot of the 659 pairs of residuals is given in Fig. 5.2 (right) and the standardized sample values of the asymmetry measures are

$$\displaystyle{3\,\mu _{\infty }(\hat{C}_{n}) \approx 0.091,\quad \sqrt{54}\,\mu _{2}(\hat{C}_{n}) \approx 0.063,\quad \sqrt{54}\,\nu _{2}(\hat{C}_{n}) \approx 0.065.}$$

Note that the small values of the asymmetry measures observed above do not come as a surprise given that the variables exhibit substantial positive correlation. For example, the sample value of Kendall’s tau is approximatively 0. 525 for the raw data and 0. 548 for the de-trended data. Whether the above sample measures of asymmetry are significantly greater than zero can only be determined using formal statistical tests, which are described next.

5.3 Testing for Symmetry

Tests of the hypothesis

$$\displaystyle{ \mathcal{H}_{0} : C = {C}^{\top } }$$

of symmetry have been developed recently by Genest et al. [23]. Their procedures are based on the rank-based statistics $R_{n} =\mu _{ 2}^{2}(\hat{C}_{n})$, $S_{n} =\nu _{ 2}^{2}(\hat{C}_{n})$ and $T_{n} =\mu _{\infty }(\hat{C}_{n})$, whose values tend to zero in probability as n → ∞ under $\mathcal{H}_{0}$. Note that the empirical copula in [23] is based on the pairs $\{(n + 1)/n\}(\hat{U}_{i},\hat{V }_{i})$. However, this slightly different definition is inconsequential asymptotically and is not adopted here.

To carry out these tests, the asymptotic null distribution of their corresponding statistic is needed. If C is regular in the sense of Definition 5.1, the so-called empirical copula process given, for all u, v ∈ [0, 1], by

$$\displaystyle{\hat{\mathbb{C}}_{n}(u,v) = \sqrt{n}\,\{\hat{C}_{n}(u,v) - C(u,v)\}}$$

converges weakly in the space ℓ[0, 1]² of bounded functions on [0, 1]² equipped with the uniform norm [41]. In other words, one has $\hat{\mathbb{C}}_{n} \rightsquigarrow \hat{ \mathbb{C}}$ as n → ∞, where $\hat{\mathbb{C}}$ is a centered Gaussian process defined, for all u, v ∈ [0, 1], by

$$\displaystyle{\hat{\mathbb{C}}(u,v) = \mathbb{C}(u,v) -\dot{C}_{1}(u,v)\mathbb{C}(u,1) -\dot{C}_{2}(u,v)\mathbb{C}(1,v).}$$

Here, $\mathbb{C}$ is a tucked C-Brownian sheet, i.e., a centered Gaussian random field with covariance function given, for all u, v, s, t ∈ [0, 1], by

$$\displaystyle{\Gamma _{\mathbb{C}}(u,v,s,t) = C(u \wedge s,v \wedge t) - C(u,v)\,C(s,t),}$$

where for arbitrary $a,b \in \mathbb{R}$, a ∧ b = min(a, b). For variants, see [16, 18, 40, 44].

Using the continuous mapping theorem, Genest et al. [23] showed that if $\hat{\mathbb{D}}_{n}$ is the empirical process defined, for all u, v ∈ [0, 1], by

$$\displaystyle{\hat{\mathbb{D}}_{n}(u,v) = \sqrt{n}\,\{\hat{C}_{n}(u,v) -\hat{ C}_{n}(v,u)\},}$$

then as n → ∞, $\hat{\mathbb{D}}_{n} \rightsquigarrow \hat{ \mathbb{D}}$ in ℓ[0, 1]², where $\hat{\mathbb{D}}$ admits the representation

$$\displaystyle{\hat{\mathbb{D}}(u,v) = \mathbb{D}(u,v) -\dot{C}_{1}(u,v)\,\mathbb{D}(u,1) -\dot{C}_{2}(u,v)\,\mathbb{D}(1,v),}$$

in terms of a centered Gaussian random field $\mathbb{D}$ with covariance function given, at each u, v, s, t ∈ [0, 1], by $\Gamma _{\mathbb{D}}(u,v,s,t) = 2\left \{\Gamma _{\mathbb{C}}(u,v,s,t) - \Gamma _{\mathbb{C}}(u,v,t,s)\right \}$. This observation leads to the following result, excerpted from [23].

Proposition 5.5.

If C is a regular symmetric copula, then as n →∞,

$$\displaystyle\begin{array}{rcl} \mathit{nR}_{n} =\int _{ 0}^{1}\int _{ 0}^{1}\{\hat{\mathbb{D}}_{ n}{(u,v)\}}^{2}\mathrm{d}v\mathrm{d}u\;\;& \rightsquigarrow & \;\;\mathbb{D}_{ R} =\int _{ 0}^{1}\int _{ 0}^{1}\{\hat{\mathbb{D}}{(u,v)\}}^{2}\mathrm{d}v\mathrm{d}u, {}\\ \mathit{nS}_{n} =\int _{ 0}^{1}\int _{ 0}^{1}\{\hat{\mathbb{D}}_{ n}{(u,v)\}}^{2}\mathrm{d}\hat{C}_{ n}(u,v)\;\;& \rightsquigarrow & \;\;\mathbb{D}_{S} =\int _{ 0}^{1}\int _{ 0}^{1}\{\hat{\mathbb{D}}{(u,v)\}}^{2}\mathrm{d}C(u,v), {}\\ {n}^{1/2}\,T_{ n} =\sup _{(u,v)\in {[0,1]}^{2}}\vert \hat{\mathbb{D}}_{n}(u,v)\vert \;\;& \rightsquigarrow & \;\;\mathbb{D}_{T} =\sup _{(u,v)\in {[0,1]}^{2}}\vert \hat{\mathbb{D}}(u,v)\vert. {}\\ \end{array}$$

Unfortunately, the limiting distribution of all three statistics depends on the underlying copula C, which is unknown.

5.3.1 p-Value Computation

As shown in [23], valid p-values for the tests based on R _n , S _n , and T _n can be computed using a bootstrap approximation based on the Multiplier Central Limit Theorem of van der Vaart and Wellner [46]; see also [4, 41] for further details. The step-by-step description of this procedure is provided below; an implementation using the R Project for Statistical Computing is available from the authors.

Step 0.:

Compute the statistic R _n , S _n or T _n .

Step 1.:

Define P _n at any u, v ∈ [0, 1] as the n ×1 vector with ith component

$$\displaystyle{P_{\mathit{in}}(u,v) =\boldsymbol{ 1}(\hat{U}_{i} \leq u,\hat{V }_{i} \leq v) -\boldsymbol{ 1}(\hat{U}_{i} \leq v,\hat{V }_{i} \leq u).}$$

Step 2.:

Fix a bandwidth ℓ _n  ∈ (0, 1 ∕ 2), typically $\ell_{n} \approx 1/\sqrt{n}$, and a large integer M. For each h ∈ { 1, …, M}, do the following.

Step 2a.:

For arbitrary v ∈ [0, 1], set

$$\displaystyle{\dot{C}_{1n}(u,v) = \left \{\begin{array}{@{}l@{\quad }l@{}} \frac{\hat{C}_{n}(2\ell_{n},v)} {2\ell_{n}} \quad &\mbox{ if }u \in [0,\ell_{n}), \\ \frac{\hat{C}_{n}(u +\ell _{n},v) -\hat{C}_{n}(u -\ell_{n},v)} {2\ell_{n}} \quad &\mbox{ if }u \in [\ell_{n},1 -\ell_{n}], \\ \frac{\hat{C}_{n}(1,v) -\hat{C}_{n}(1 - 2\ell_{n},v)} {2\ell_{n}} \quad &\mbox{ if }u \in (1 -\ell_{n},1]. \end{array} \right.}$$

Similarly, for arbitrary u ∈ [0, 1], set

$$\displaystyle{\dot{C}_{2n}(u,v) = \left \{\begin{array}{@{}l@{\quad }l@{}} \frac{\hat{C}_{n}(u,2\ell_{n})} {2\ell_{n}} \quad &\mbox{ if }v \in [0,\ell_{n}), \\ \frac{\hat{C}_{n}(u,v +\ell _{n}) -\hat{C}_{n}(u,v -\ell_{n})} {2\ell_{n}} \quad &\mbox{ if }v \in [\ell_{n},1 -\ell_{n}], \\ \frac{\hat{C}_{n}(u,1) -\hat{C}_{n}(u,1 - 2\ell_{n})} {2\ell_{n}} \quad &\mbox{ if }v \in (1 -\ell_{n},1]. \end{array} \right.}$$

Step 2b.:

Draw a vector ${\xi }^{(h)} = (\xi _{1}^{(h)},\ldots ,\xi _{n}^{(h)})$ of independent nonnegative random variables with unit mean and unit variance; the standard exponential distribution is typically used to this end. Set

$$\displaystyle{\bar{\xi }_{n}^{(h)} = \frac{1} {n}\,(\xi _{1}^{(h)} + \cdots +\xi _{ n}^{(h)})\quad \text{and}\quad \Xi _{ n}^{(h)} = \left ( \frac{\xi _{1}^{(h)}} {\bar{\xi }_{n}^{(h)}} - 1,\ldots , \frac{\xi _{n}^{(h)}} {\bar{\xi }_{n}^{(h)}} - 1\right ).}$$

Step 2c.:

Define the bootstrap replicate $\hat{\mathbb{D}}_{n}^{(h)}$ of $\hat{\mathbb{D}}$ at any u, v ∈ [0, 1] by

$$\displaystyle{\hat{\mathbb{D}}_{n}^{(h)}(u,v) = \frac{1} {\sqrt{n}}\Xi _{n}^{(h)}\{P_{ n}(u,v) -\dot{C}_{1n}(u,v)P_{n}(u,1) -\dot{C}_{2n}(u,v)P_{n}(1,v)\}.}$$

Step 2d.:

Compute the bootstrap replicate of the appropriate test statistic, viz.

$$\displaystyle\begin{array}{rcl} R_{n}^{(h)}& =& \frac{1} {n}\int _{0}^{1}\int _{ 0}^{1}\{\hat{\mathbb{D}}_{ n}^{(h)}{(u,v)\}}^{2}\mathrm{d}v\mathrm{d}u, {}\\ S_{n}^{(h)}& =& \frac{1} {n}\int _{0}^{1}\int _{ 0}^{1}\{\hat{\mathbb{D}}_{ n}^{(h)}{(u,v)\}}^{2}\mathrm{d}\hat{C}_{ n}(u,v), {}\\ T_{n}^{(h)}& =& \frac{1} {\sqrt{n}}\sup _{(u,v)\in {[0,1]}^{2}}\vert \hat{\mathbb{D}}_{n}^{(h)}(u,v)\vert. {}\\ \end{array}$$

Step 3.:

Compute the approximate p-value, viz.

$$\displaystyle{ \frac{1} {M}\sum _{h=1}^{M}\boldsymbol{1}(R_{ n}^{(h)} > R_{ n}),\quad \frac{1} {M}\sum _{h=1}^{M}\boldsymbol{1}(S_{ n}^{(h)} > S_{ n}),\quad \frac{1} {M}\sum _{h=1}^{M}\boldsymbol{1}(T_{ n}^{(h)} >\, T_{ n}).}$$

It is easy to compute $S_{n}^{(h)}$ because $\hat{C}_{n}$ is a discrete distribution function. For the other two statistics, the computational burden can be reduced by resorting to a numerical approximation involving an N ×N grid, viz.

$$\displaystyle\begin{array}{rcl} R_{n}^{(h)}& \approx & \frac{1} {{\mathit{nN}}^{2}}\sum _{k=1}^{N}\sum _{ \ell=1}^{N}{\left \{\hat{\mathbb{D}}_{ n}^{(h)}\left ( \frac{k} {N}\,, \frac{\ell} {N}\right )\right \}}^{2},{}\end{array}$$

(5.4a)

$$\displaystyle\begin{array}{rcl} T_{n}^{(h)}& \approx & \frac{1} {\sqrt{n}}\,\max _{k,\ell\in \{1,\ldots ,N\}}\left \vert \hat{\mathbb{D}}_{n}^{(h)}\left ( \frac{k} {N}\,, \frac{\ell} {N}\right )\right \vert.{}\end{array}$$

(5.4b)

The results of an extensive Monte Carlo simulation study comparing the power of these three tests were reported in [23]. The test based on the Cramér—von Mises statistic S _n was found to be generally more powerful than its competitors. It is also the quickest to perform and hence can be recommended on that account too.

5.3.2 Illustration

Consider once again the monthly average flow rate of the Inn and Danube rivers. The tests based on R _n , S _n , and T _n were applied to both the raw and de-trended data using M = 1, 000 multiplier replicates, a bandwidth $\ell_{n} = 0.04 \approx 1/\sqrt{660}$ and N = 100 grid points in the approximation (5.4). The hypothesis $\mathcal{H}_{0}$ of symmetry was rejected in all cases. The p-values were essentially zero except when the statistic T _n was used on the de-trended data, where it was found that p ≈ 1. 5 %.

5.4 Asymmetric Copula Families

When the null hypothesis of symmetry is rejected, dependence models based on Archimedean and meta-elliptical copula families are ruled out straightaway, because they cannot account for asymmetry; popular alternatives are reviewed below.

5.4.1 Extreme-Value Copulas

A copula C is said to be of the extreme-value type if and only if there exists a function A : [0, 1] → [1 ∕ 2, 1] such that, for all u, v ∈ [0, 1],

$$\displaystyle{ C(u,v) =\exp \left [\ln (uv)A\left \{ \frac{\ln (v)} {\ln (uv)}\right \}\right ]. }$$

(5.5)

For C to be a copula, the so-called Pickands dependence function A must be convex and such that, for all t ∈ [0, 1], max(t, 1 − t) ≤ A(t) ≤ 1. The bounds A(t) ≡ 1 and $A(t) =\max (t,1 - t)$ correspond to the independence copula and the Fréchet–Hoeffding upper bound, respectively.

An extreme-value copula is asymmetric if and only if its Pickands dependence function A is asymmetric with respect to 1 ∕ 2, i.e., if there exists t ∈ [0, 1] such that A(t)≠A(1 − t). Klement and Mesiar [28] state that if C is of the form (5.5), then

$$\displaystyle{\mu _{\infty }(C) \leq {4}^{4}/{5}^{5} \approx 0.082.}$$

A detailed proof of this result is given by Durante and Mesiar [9], who show that the upper bound is reached for two members of the Marshall–Olkin extreme-value copula family whose Pickands dependence functions are given, for all t ∈ [0, 1], by

$$\displaystyle{ A_{1}(t) =\max \left (1 - t, \frac{t + 1} {2} \right ),\quad A_{2}(t) =\max \left (t, \frac{2 - t} {2} \right ). }$$

(5.6)

Typical examples of symmetric extreme-value copulas include the Galambos, Gumbel, Hüsler–Reiß, Tawn, and t-EV families; see [19] for their definitions and further details. Each of these families can be made asymmetric using Khoudraji’s device [21, 27]. The latter is based on the observation that if C is an extreme-value copula with Pickands dependence function A and $\lambda ,\kappa \in (0,1)$, then the copula given, for all u, v ∈ [0, 1], by

$$\displaystyle{ C_{\lambda ,\kappa }(u,v) = {u}^{1-\lambda }{v}^{1-\kappa }C({u}^{\lambda },{v}^{\kappa }) }$$

(5.7)

is again extreme-value with Pickands dependence function of the form

$$\displaystyle{A_{\lambda ,\kappa }(t) = (1-\kappa )t + (1-\lambda )(1 - t) +\{\kappa t +\lambda (1 - t)\}A\left \{ \frac{\kappa t} {\kappa t +\lambda (1 - t)}\right \}.}$$

A random pair (U, V ) from $C_{\lambda ,\kappa }$ can be obtained as follows.

Step 1.:

Draw independent random variables W and Z, uniform on [0, 1].

Step 2.:

Draw a pair (X, Y ) from the copula C and set

$$\displaystyle{U =\max ({W}^{1/(1-\lambda )},{X}^{1/\lambda }),\quad V =\max ({Z}^{1/(1-\kappa )},{Y }^{1/\kappa }).}$$

Clearly, $C_{\lambda ,\kappa }$ is asymmetric when $\lambda \neq \kappa$. Figure 5.4 shows the effect of Khoudraji’s device when C is a Gumbel copula with τ ∈ { 0. 5, 0. 75, 0. 9} and $\lambda = 0.5,\kappa = 0.7$; samples from the symmetric Gumbel copula and from $C_{\lambda ,\kappa }$ are displayed in the top and middle row, respectively. The plots confirm that asymmetry restricts the range of attainable positive association, as observed in Sect. 5.2.1. It is shown in [22] that Kendall’s tau of $C_{\lambda ,\kappa }$ satisfies

$$\displaystyle{ \tau (C_{\lambda ,\kappa }) \leq \frac{\kappa \lambda } {\kappa +\lambda -\kappa \lambda } =\tau _{\max }(\lambda ,\kappa ). }$$

(5.8)

Khoudraji’s device can be generalized by choosing an integer m ≥ 1, extreme-value copulas $C_{1},\ldots ,C_{m}$, and vectors $\boldsymbol{\lambda }= (\lambda _{1},\ldots ,\lambda _{m}),\boldsymbol{\kappa }= (\kappa _{1},\ldots ,\kappa _{m}) \in {[0,1]}^{m}$ whose components sum up to 1. The copula defined, for all u, v ∈ [0, 1], by

$$\displaystyle{ C_{\boldsymbol{\lambda },\boldsymbol{\kappa }}(u,v) =\prod _{ j=1}^{m}C_{ j}({u}^{\lambda _{j} },{v}^{\kappa _{j}}) }$$

(5.9)

is then an extreme-value copula with Pickands dependence function

$$\displaystyle{A_{\boldsymbol{\lambda },\boldsymbol{\kappa }}(t) =\sum _{ j=1}^{m}\{\kappa _{ j}t +\lambda _{j}(1 - t)\}A_{j}\left \{ \frac{\kappa _{j}t} {\kappa _{j}t +\lambda _{j}(1 - t)}\right \}.}$$

This result is proved, e.g., by Bacigál et al. [2], who mention other constructions.

Fig. 5.4

Top row: samples of size 2, 000 from the Gumbel copula with τ = 0. 5 (left), τ = 0. 75 (middle) and τ = 0. 9 (right). The middle and bottom row show the effect of asymmetrization of these copulas using Khoudraji’s device with λ = 0. 5 and κ = 0. 7 and the Liouville copula construction with α = 1 and β = 20, respectively

The availability of statistical tools for model fitting and validation makes extreme-value copulas particularly convenient. However, these dependence structures may not always be appropriate. In particular, copulas of the form (5.5) are stochastically increasing [17] and exhibit upper-tail but no lower-tail dependence.

5.4.2 Asymmetric Generalizations of Archimedean Copulas

A copula C is called Archimedean if it can be written, for all u, v ∈ [0, 1], in the form

$$\displaystyle{C(u,v) {=\varphi }^{-1}\{\varphi (u) +\varphi (v)\}}$$

in terms of a strictly decreasing, convex map $\varphi : (0,1] \rightarrow [0,\infty )$ such that $\varphi (1) = 0$. By convention, $\varphi (0) =\lim _{u\downarrow 0}\varphi (u)$ and ${\varphi }^{-1}(s) = 0$ when $s \geq \varphi (0)$. The function $\varphi$ is referred to as an Archimedean generator; see Chap. 4 in [36] for examples.

Clearly, any Archimedean copula is symmetric. However, Khoudraji’s device can again be used to generate an asymmetric copula from an Archimedean copula C with generator $\varphi$ through (5.7). The resulting copula $C_{\lambda ,\kappa }$ is no longer Archimedean; see [34] for further details and illustrations.

Although Khoudraji’s device induces asymmetry, it does not provide much freedom in modeling association between the variables. To see this, observe that for given $\lambda ,\kappa$, the Fréchet–Hoeffding inequality implies that, for all u, v ∈ [0, 1],

$$\displaystyle{\max (0,u{v}^{1-\kappa } + {u}^{1-\lambda }v - {u}^{1-\lambda }{v}^{1-\lambda }) \leq C_{\lambda ,\kappa }(u,v) \leq \min (u{v}^{1-\kappa },{u}^{1-\lambda }v).}$$

The attainable range of values of $\tau (C_{\lambda ,\kappa })$ is thus $[\tau _{\min }(\lambda ,\kappa ),\tau _{\max }(\lambda ,\kappa )]$ with $\tau _{\max }(C_{\lambda ,\kappa })$ given in (5.8) and

$$\displaystyle{\tau _{\min }(\lambda ,\kappa ) = \frac{\kappa \lambda } {\kappa +\lambda -\kappa \lambda }\,\mathrm{B}\left ( \frac{\lambda } {2} - 1, \frac{\kappa } {2} - 1\right ) - \frac{2\kappa \lambda } {\lambda +\kappa -\lambda \kappa },}$$

where B(x, y), x, y > 0, denotes the Beta function. The length of this interval, $\tau _{\max }(\lambda ,\kappa ) -\tau _{\min }(\lambda ,\kappa )$, is shown in Fig. 5.3. For example, it can be seen that $\tau (C_{\lambda ,\kappa })$ is small when $\vert \lambda -\kappa \vert$ is large. Although the precise interplay between the parameters $\lambda ,\kappa$ and the asymmetry measures presented in Sect. 5.2 is not known at present, asymmetry appears to increase with $\vert \lambda -\kappa \vert$.

An alternative asymmetric generalization of Archimedean copulas has recently been proposed by McNeil and Nešlehová [33]. Using the fact that Archimedean copulas are survival copulas of simplex distributions [32], one can consider survival copulas of the more general class of Liouville distributions. A random pair (X, Y ) is said to follow a Liouville distribution if $(X,Y ){ \text{d} \atop =} R \times (D_{1},D_{2})$, where ${ \text{d} \atop =}$ denotes equality in distribution, R is a strictly positive random variable independent of the random pair $(D_{1},D_{2})$ having Dirichlet$(\alpha ,\beta )$ distribution with α, β > 0. When $\alpha =\beta = 1$, (X, Y ) follows a simplex distribution and its survival copula is Archimedean. The interesting case arises when α≠β, as the survival copula is then asymmetric.

Although Liouville copulas do not have a closed form in general, the expression for their density as well as random number generation is comparatively tractable when α, β are integer-valued and the distribution of R is suitably chosen.

When the inverse generator $\psi {=\varphi }^{-1}$ is completely monotone, one option is to set $R{ \text{d} \atop =} \varphi (W_{1}) + \cdots +\varphi (W_{\alpha +\beta })$, where the distribution of $(W_{1},\ldots ,W_{\alpha +\beta })$ is an $(\alpha +\beta )$-dimensional Archimedean copula $C_{\varphi }$ with generator $\varphi$. A random pair (U, V ) from the corresponding Liouville copula with parameters α, β is then obtained as follows.

Step 1.:

Draw $(W_{1},\ldots ,W_{\alpha +\beta })$ from the multivariate Archimedean copula $C_{\varphi }$.

Step 2.:

Set $X =\varphi (W_{1}) + \cdots +\varphi (W_{\alpha })$ and $Y =\varphi (W_{\alpha +1}) + \cdots +\varphi (W_{\alpha +\beta })$.

Step 3.:

For $j = 1,\ldots ,\alpha \vee \beta - 1$, compute the jth derivative ψ ^(j) of ψ and set

$$\displaystyle{U =\sum _{ j=0}^{\alpha -1}{(-1)}^{j}{\frac{{X}^{j}} {j!} \psi }^{(j)}(X),\quad V =\sum _{ j=0}^{\beta -1}{(-1)}^{j}{\frac{{Y }^{j}} {j!} \psi }^{(j)}(Y ).}$$

The bottom row of Fig. 5.4 shows samples of size 2, 000 from Liouville copulas C _α, β with α = 1 and β = 20 when $C_{\varphi }$ is Gumbel’s copula with τ ∈ { 0. 5, 0. 75, 0. 9}. The high-order derivatives of the Gumbel generator were computed using Theorem 2 in [25]. For lack of a theoretical expression, α and β were rigged so that the sample values of $\mu _{2}(C_{\alpha ,\beta })$ and $\mu _{2}(C_{\lambda ,\kappa })$ are close when τ = 0. 5. It transpires from the plots that in contrast to Khoudraji’s device, the Liouville construction does not restrict the range of attainable association. This was demonstrated more formally in [33], where an explicit but cumbersome formula for Kendall’s tau may be found.

5.4.3 Archimax Copulas

Archimax copulas provide yet another class of asymmetric bivariate copulas. Following Capéraà et al. [5], a copula C belongs to this class if, for all u, v ∈ (0, 1),

$$\displaystyle{ C(u,v) {=\varphi }^{-1}\left [\{\varphi (u) +\varphi (v)\} \times A\left \{ \frac{\varphi (v)} {\varphi (u) +\varphi (v)}\right \}\right ], }$$

(5.10)

where A is a Pickands dependence function and $\varphi$ is a bivariate Archimedean generator. The class is called Archimax because:

• (a) If A ≡ 1, then (5.10) reduces to an (exchangeable) Archimedean copula.

• (b) If $\varphi (t) = -\ln (t)$ for all t ∈ (0, 1], then (5.10) is an extreme-value copula.

An Archimax copula C with generator $\varphi$ and Pickands dependence function A is symmetric when $A(t) = A(1 - t)$ for all t ∈ [0, 1]. To construct an asymmetric Archimax copula, one can thus resort to any of the techniques described in Sect. 5.4.1. Following [5, 24], a random pair (U, V ) from C can be obtained as follows.

Step 1.:

Draw a pair (U ₁, V ₁) from an extreme-value copula with Pickands dependence function A and set $Z =\ln (U_{1})/\ln (U_{1}V _{1})$.

Step 2.:

Compute the first and second order derivatives A ^′ and A ^{′ ′} of A and set

$$\displaystyle{p(Z) = \frac{Z(1 - Z){A}^{{\prime\prime}}(Z)} {A(Z) + (1 - 2Z){A}^{{\prime}}(Z) + Z(1 - Z)[{A}^{{\prime\prime}}(Z) -\{ {A}^{{\prime}}{(Z)\}}^{2}/A(Z)]}.}$$

Step 3.:

Generate a pair $(U_{2},V _{2})$ from an Archimedean copula $C_{\varphi }$ with generator $\varphi$ and a random variable U ₃ uniformly distributed on (0, 1).

Step 4.:

Set W = U ₂ if U ₃ ≤ p(Z) and $W = C_{\varphi }(U_{2},V _{2})$ otherwise and compute

$$\displaystyle{U {=\varphi }^{-1}\left \{\frac{Z\varphi (W)} {A(Z)} \right \},\quad V {=\varphi }^{-1}\left \{\frac{(1 - Z)\varphi (W)} {A(Z)} \right \}.}$$

Fig. 5.5

Samples of size 2, 000 from the Archimax copula with Gumbel’s asymmetric Pickands dependence function with λ = 0. 5, κ = 0. 7 and θ = 2 (left), θ = 4 (middle) and θ = 10 (right). The Archimedean generator $\varphi$ is the Clayton (top row), Frank (middle row), and Joe (bottom row); its parameter is chosen so that $\tau (C_{\varphi }) = 0.5$

Typical samples from Archimax copulas are shown in Fig. 5.5. As can be seen, the degree of asymmetry is modest. This is consistent with the finding of Durante and Mesiar [9] who show that

$$\displaystyle{ \mu _{\infty }(C) \leq \sup _{t\in (0,\infty )}{\vert \varphi }^{-1}(t) {-\varphi }^{-1}(5t/4)\vert. }$$

(5.11)

Once again, the upper bound is reached if either A = A ₁ or A = A ₂, as defined in (5.6). For the Clayton, Frank and Joe copulas used in Fig. 5.5, the function ${\varphi }^{-1}(t) {-\varphi }^{-1}(5t/4)$ is displayed in Fig. 5.6. The upper bound on μ _∞ in (5.11) is shown in the right panel of the same figure. One can see that the level of attainable asymmetry decreases rather quickly with increasing τ.

Fig. 5.6

Left panel: The function ${\varphi }^{-1}(t) {-\varphi }^{-1}(5t/4)$ for the Clayton (full), Frank (dashed), and Joe (dotted) copulas with τ = 0. 5. Right panel: The upper bound on μ _∞ from (5.11) as a function of τ for the Clayton (full), Frank (dashed) and Joe (dotted) copulas

5.4.4 Algebraic Constructions

In recent years, various other ways of constructing asymmetric copulas have been proposed. They are merely outlined below as in most cases, comparatively little is known about them, particularly from a practical perspective.

As was already apparent from [21], different extensions of Khoudraji’s device are possible. Two of them have been investigated in detail. Given m ≥ 2 copulas $C_{1},\ldots ,C_{m}$, Liebscher [30, 31] considers copulas defined, for all u, v ∈ [0, 1], by

$$\displaystyle{C(u,v) =\prod _{ j=1}^{m}C_{ j}\{f_{j}(u),g_{j}(v)\},}$$

where $f_{1},\ldots ,f_{m},g_{1},\ldots ,g_{m}$ are strictly increasing mappings from [0, 1] to [0, 1] such that, for all t ∈ [0, 1], $f_{1}(t) \times \cdots \times f_{m}(t) = g_{1}(t) \times \cdots \times g_{m}(t) = t$. A simple procedure for generating observations from such copulas is given in [30, 31]. However, no practical guidance for the choice of the functions is provided.

Note that (5.9) corresponds to the case where $f_{j}(t) = {t}^{\lambda _{j}}$ and $g_{j}(t) = {t}^{\kappa _{j}}$ for all j ∈ { 1, …, m}. When $m = 2$, taking $f_{2}(t) = t/f_{1}(t)$ and $g_{2}(t) = t/g_{1}(t)$ for all t ∈ (0, 1] leads to a construction proposed independently by Durante [7]. In his paper, this author also considers mappings defined, for all u, v ∈ [0, 1], by

$$\displaystyle{ C(u,v) = C_{3}[C_{1}\{f_{1}(u),g_{1}(v)\},C_{2}\{f_{2}(u),g_{2}(v)\}] }$$

(5.12)

in terms of fixed copulas C ₁, C ₂, and C ₃. Further suppose that C ₃ is convex in each variable, so that if (U, V ) has distribution C ₃, U is stochastically decreasing in V and vice versa. Then the function C in (5.12) is a copula provided that for all u ∈ [0, 1],

$$\displaystyle{C_{3}\{f_{1}(u),f_{2}(u)\} = C_{3}\{g_{1}(u),g_{2}(u)\} = u.}$$

Even greater generality is envisaged in [7] by relaxing the conditions on C _i , i = 1, 2, 3, but until rich examples, probabilistic interpretations, and simulation algorithms have been found, this approach remains somewhat of an empty vessel.

The work of Durante et al. [12] is of much more practical interest. Given a bivariate Archimedean generator $\varphi$, they define a copula C, for all u, v ∈ (0, 1), by

$$\displaystyle{C(u,v) = u\left [1 {-\varphi }^{-1}\left \{\varphi (1 - v)/u\right \}\right ].}$$

They prove that C is positive quadrant dependent and that it is symmetric if and only if it is the Fréchet–Hoeffding upper bound or a Clayton copula with parameter θ > 0. To obtain the latter, Durante and Jaworski [8] show that one must take $\varphi (t) =\{ {(1 - t)}^{-\theta }- {1\}}^{-1/\theta }$ for all t ∈ (0, 1). It is also easy to simulate from C as follows.

Step 1.:

Draw a pair (Z, V ) from an Archimedean copula with generator $\varphi$.

Step 2.:

Return (U, 1 − V ), where $U =\varphi (V )/\{\varphi (Z) +\varphi (V )\}$.

An illustration of this asymmetrization technique is provided in Fig. 5.7 for three Gumbel copula generators corresponding to different degrees of dependence. The plots suggest that the degree of asymmetry decreases as τ increases. In [12], constructions leading to negative quadrant dependence are also discussed; they are obtained upon considering the copula given by $u - C(u,1 - v) = {u\varphi }^{-1}\{\varphi (v)/u\}$.

Fig. 5.7

Samples of size 2, 000 from the Durante–Jaworski–Mesiar asymmetrization of the Gumbel copula with τ = 0. 1 (left), τ = 0. 5 (middle) and τ = 0. 9 (right)

Yet another approach is taken by Alfonsi and Brigo [1]. They consider absolutely continuous copulas whose density is expressed, at every u, v ∈ [0, 1], by $c(u,v) =\ell (u - v)$ in terms of some function ℓ : [ − 1, 1] → [0, ∞). For this construction to be valid, the authors must assume that ℓ is twice finitely integrable, that $\int _{0}^{1}\ell(t)\mathrm{d}t = 1$, and that $\ell(t) =\ell (t - 1)$ for all t ∈ [0, 1]. Letting $L(t) =\int _{ 0}^{t}\int _{-1}^{s}\ell(w)\mathrm{d}w\mathrm{d}s$, it can then be shown that C is indeed a copula and that, for all u, v ∈ [0, 1],

$$\displaystyle{C_{\ell}(u,v) = L(u) + L(-v) - L(u - v).}$$

This construction is of little interest, however, because if (U, V ) has copula C _ℓ , the pair (U, 1 − V ) then has a symmetric copula. Indeed, the conditions on ℓ imply that for all t ∈ [0, 1], $L(t) = L(1) + L(t - 1) - (1 - t)$ and hence, for all u, v ∈ [0, 1], $u - C_{\ell}(u,1 - v) = v - C_{\ell}(v,1 - u)$. Thus for C _ℓ to be a realistic copula of (X, Y ), the dependence structure of (X, − Y ) must be symmetric. Given the wealth of such models, a statistician would be much better off analyzing the latter pair.

Finally, asymmetric copulas could also be constructed by gluing copulas or using ordinal sums; see, e.g., [35, 42]. For asymmetric copulas with given diagonal section, see [6, 14, 15, 38].

5.5 Discussion

Returning to the hydrological data, it has already been seen that both the raw and de-trended data exhibit asymmetry. As it makes more sense to model pure dependence, this discussion concentrates on the analysis of the de-trended data.

While Bacigál et al. [2] consider Archimax models for these data, the hypothesis that the copula is of the simpler extreme-value form (5.5) cannot be rejected. For example, the rank-based tests developed in [3] and [29] yield approximate p-values of 96.8 % and 7.34 %, respectively.

In order to choose a suitable parametric copula family, asymmetric Galambos, Gumbel, Hüsler–Reiß and Tawn extreme-value copulas of the form (5.7) were fitted to the data by the maximum pseudo likelihood method of Genest et al. [20]. The four models returned very similar results; the highest likelihood was obtained for the Hüsler–Reiß with parameters $\hat{\theta }= 2.16$, $\hat{\lambda }= 0.938$ and $\hat{\kappa }= 1$. At 283. 23, the corresponding log-likelihood is also the highest compared to the models in [2].

The parametric estimate of the Pickands dependence function of the asymmetric Hüsler–Reiß copula is shown in the left panel of Fig. 5.8. Rank-based versions of the nonparametric estimates of A due to Pickands (P) and Capéraà, Fougères, and Genest (CFG) are also displayed for comparison purposes. The right panel displays a random sample of size 659 from the fitted asymmetric Hüsler–Reiß model. Visual comparison with the rank plot of the original data suggests an adequate fit. This could be checked formally using the goodness-of-fit procedures introduced in [22].

Fig. 5.8

Left panel: parametric Pickands dependence function estimated from the de-trended data using the asymmetric Hüsler–Reiß model (solid line), together with the rank-based nonparametric P (dotted line) and CFG (dashed line) estimators. Right panel: a sample of size 659 from the fitted asymmetric Hüsler–Reiß copula

In contrast, the hypothesis of extremeness is clearly rejected for the raw data. If one were to model the latter (i.e., deliberately ignoring the influence of time), one would need to resort to more elaborate dependence models, such as the Archimax, Liouville or Durante–Jaworski–Mesiar copulas. While these constructions clearly have potential, practical tools for their implementation remain to be developed.

5.6 Appendix

Proof of Proposition 5.3. To determine the limit of $\nu _{p}(\hat{C}_{n})$, proceed as in the proof of Proposition 4 in Genest et al. [23]. Write

$$\displaystyle{\vert \nu _{p}^{p}(\hat{C}_{ n}) -\nu _{p}^{p}(C)\vert \leq \vert \alpha _{ n}\vert + \vert \beta _{n}\vert ,}$$

where

$$\displaystyle{\alpha _{n}\,=\!\int _{0}^{1}\!\int _{ 0}^{1}\!\{\hat{C}_{ n}(u,v) -\hat{ C}_{n}{(v,u)\}}^{p}\mathrm{d}\hat{C}_{ n}(u,v) -\int _{0}^{1}\!\int _{ 0}^{1}\!{\left \{C(u,v) - C(v,u)\right \}}^{p}\mathrm{d}\hat{C}_{ n}(u,v)}$$

and

$$\displaystyle{\beta _{n}\,=\!\int _{0}^{1}\!\int _{ 0}^{1}\!{\left \{C(u,v) - C(v,u)\right \}}^{p}\mathrm{d}\hat{C}_{ n}(u,v) -\int _{0}^{1}\!\int _{ 0}^{1}\!{\left \{C(u,v) - C(v,u)\right \}}^{p}\mathrm{d}C(u,v).}$$

By the Mean Value Theorem, $\vert {a}^{p} - {b}^{p}\vert \leq p\,{2}^{p-1}\vert a - b\vert$ for all a, b ∈ [0, 2]. Because | C(u, v) − C(v, u) | and $\vert \hat{C}_{n}(u,v) -\hat{ C}_{n}(v,u)\vert$ take values in [0, 2] for all u, v ∈ [0, 1],

$$\displaystyle\begin{array}{rcl} & & \left \vert \vert \hat{C}_{n}(u,v) -\hat{ C}_{n}(v,u){\vert }^{p} -\vert C(u,v) - C(v,u){\vert }^{p}\right \vert {}\\ & \leq & p\,{2}^{p-1}\left \vert \vert \hat{C}_{ n}(u,v) -\hat{ C}_{n}(v,u)\vert -\vert C(u,v) - C(v,u)\vert \right \vert {}\\ & \leq & p\,{2}^{p}\sup _{ u,v\in [0,1]}\vert \hat{C}_{n}(u,v) - C(u,v)\vert , {}\\ \end{array}$$

where the last step follows from a twofold application of the triangular inequality. In particular, therefore,

$$\displaystyle{\vert \alpha _{n}\vert \leq p\,{2}^{p}\sup _{ u,v\in [0,1]}\vert \hat{C}_{n}(u,v) - C(u,v)\vert }$$

and hence tends to 0 in probability, as n → ∞. Turning to β _n , set $\psi = {(C - {C}^{\top })}^{p}$ and apply Proposition A.1(i) in [20], taking δ = 1 therein. It then follows that

$$\displaystyle{\int _{0}^{1}\int _{ 0}^{1}{\left \{C(u,v) - C(v,u)\right \}}^{p}\mathrm{d}\hat{C}_{ n}(u,v) = \frac{1} {n}\sum _{i=1}^{n}\psi \left (\frac{R_{i}} {n} , \frac{S_{i}} {n} \right ) \rightarrow \int _{0}^{1}\int _{ 0}^{1}\psi (u,v)\mathrm{d}C(u,v)}$$

almost surely, whence | β _n  | converges to 0 in probability, as n → ∞. □ 

Proof of Proposition 5.4. The fact that ν _p satisfies axioms (B1), (B3), and (B4) is easily seen. It is also immediate that ν _p (C) = 0 if C is symmetric. To establish the converse, assume for simplicity that C has a density c. If ν _p (C) = 0, then the supports of C and ${C}^{\top }$ are contained in $A =\{ (u,v) : C(u,v) = C(v,u)\}$. Given that c(u, v) = c(v, u) on A, it follows that

$$\displaystyle{C(u,v) =\int _{ 0}^{u}\int _{ 0}^{v}\boldsymbol{1}\{(s,t) \in A\}c(s,t)\mathrm{d}t\mathrm{d}s =\int _{ 0}^{u}\int _{ 0}^{v}\boldsymbol{1}\{(s,t) \in A\}c(t,s)\mathrm{d}t\mathrm{d}s = C(v,u).}$$

Regarding axiom (B5), write, in analogy with the proof of Proposition 5.3,

$$\displaystyle{\vert \nu _{p}^{p}(C_{ n}) -\nu _{p}^{p}(C)\vert \leq \vert \alpha _{ n}\vert + \vert \beta _{n}\vert ,}$$

where

$$\displaystyle{\alpha _{n} =\int _{ 0}^{1}\int _{ 0}^{1}{\left \vert C_{ n}(u,v) - C_{n}(v,u)\right \vert }^{p}\mathrm{d}C_{ n}(u,v) -\int _{0}^{1}\int _{ 0}^{1}{\left \vert C(u,v) - C(v,u)\right \vert }^{p}\mathrm{d}C_{ n}(u,v)}$$

and

$$\displaystyle{\beta _{n} =\int _{ 0}^{1}\int _{ 0}^{1}{\left \vert C(u,v) - C(v,u)\right \vert }^{p}\mathrm{d}C_{ n}(u,v) -\int _{0}^{1}\int _{ 0}^{1}{\left \vert C(u,v) - C(v,u)\right \vert }^{p}\mathrm{d}C(u,v).}$$

By the same argument as in the proof of Proposition 5.3,

$$\displaystyle{\vert \alpha _{n}\vert \leq p{2}^{p}\sup _{ u,v\in [0,1]}\vert C_{n}(u,v) - C(u,v)\vert }$$

and hence α _n  → 0 as n → ∞. The fact that β _n  → 0 follows directly from the weak convergence of C _n to C; see, e.g., Lemma 2.2 in [45]. □