On comprehensive families of copulas involving the three basic copulas and transformations thereof
-
Susanne Saminger-Platz
,
Anna Kolesárová
,
Adam Šeliga
,
Radko Mesiar
and
Erich Peter Klement
Abstract
Comprehensive families of copulas including the three basic copulas (at least as limit cases) are useful tools to model countermonotonicity, independence, and comonotonicity of pairs of random variables on the same probability space. In this contribution, we study how the transition from a (basic) copula to a copula modeling a different dependence behavior can be realized by means of ordinal sums based on one of the three basic copulas, perturbing one of the three basic copulas (considering some appropriate parameterized transformations) and truncating the results using the Fréchet-Hoeffding bounds. We provide results and examples showing the flexibility and the restrictions for obtaining new copulas or comprehensive families and illustrate the development of their dependence parameters.
1 Introduction and motivation
Copulas are functions that link a multivariate distribution function with its one-dimensional margins. Thanks to Sklar’s theorem [22,38,65,76], we know that, whenever
The lower Fréchet-Hoeffding bound
These copulas, often called the three basic copulas, capture – in the language of Sklar’s theorem – the countermonotonicity, the independence, and the comonotonicity of pairs of random variables on the same probability space. Moreover, the Fréchet-Hoeffding lower and upper bounds
Following Nelsen [65, p. 15], families of copulas including the three basic copulas
There are several construction methods determining new copulas from two or more given copulas, e.g., using additive or multiplicative generators (leading to the so-called Archimedean copulas that are isomorphic to either
In the literature, there is an abundance of comprehensive families of distinguished copulas: well-known representatives (see, e.g., [45,65]) are convex combinations of the three basic copulas
In this article, we will expand some ideas mentioned in, e.g., [12,37,41,49,69,71], and investigate parameterized transformations of copulas (as given in Definition 2.7) with or without truncation (2.2) as well as ordinal sums based on one of the three basic copulas (Definition 2.5), and study the dependence behavior of many of these constructions. Note that some of the parameterized transformations discussed here are related to the construction of copulas by means of pairs of order statistics [12] or to quadratic constructions of copulas [49]. Others may be seen as generalizations of the family of Eyraud-Farlie-Gumbel Morgenstern copulas (see, e.g., [65, Example 3.12]) or its family extension as discussed by Hürlimann in [37] (compare also [71, Example 2.12(ii)]), or as perturbations of copulas (compare also [13,62,69–71,74]).
We start off and use only the Fréchet-Hoeffding bounds
The structure of this article is as follows: after giving the necessary preliminaries in Section 2, we take a closer look at M- and W-ordinal sums of transformed copulas (Section 3). In Section 4, we provide an in-depth study of
2 Preliminaries
After some preliminary work by Fréchet [25] on the set of
Definition 2.1
A function
From properties [C1] and [C2], it follows immediately that each copula
Definition 2.2
We also consider two larger classes of bivariate functions, namely,
the set
Definition 2.3
For an arbitrary function
We consider also the truncation
It is straightforward that for each function
In the case of a copula
is also a copula, called the survival copula
and for each
Remark 2.4
Theorem 3.1.2 in [22] tells us that each bivariate copula
![Figure 1
Values of the copula
C
C
, the survival copula
C
surv
{C}^{{\rm{surv}}}
, the
x
x
-flipping
C
x
flip
{C}^{x{\rm{flip}}}
, and the
y
y
-flipping
C
y
flip
{C}^{y{\rm{flip}}}
at the red point
(
u
,
v
)
\left(u,v)
are given by the
P
C
{P}_{C}
-measure of the rectangle in yellow, cyan, green, and magenta, respectively, i.e., we obtain
C
(
u
,
v
)
=
P
C
(
[
0
,
u
]
×
[
0
,
v
]
)
C\left(u,v)={P}_{C}\left(\left[0,u]\times \left[0,v])
(yellow),
C
surv
(
u
,
v
)
=
P
C
(
[
1
‒
u
,
1
]
×
[
1
‒
v
,
1
]
)
{C}^{{\rm{surv}}}\left(u,v)={P}_{C}\left({[}1‒u,1]\times {[}1‒v,1])
(cyan),
C
x
flip
(
u
,
v
)
=
P
C
(
[
1
‒
u
,
1
]
×
[
0
,
v
]
)
{C}^{x{\rm{flip}}}\left(u,v)={P}_{C}\left(\left[1‒u,1]\times \left[0,v])
(green), and
C
y
flip
(
u
,
v
)
=
P
C
(
[
0
,
u
]
×
[
1
‒
v
,
1
]
)
{C}^{y{\rm{flip}}}\left(u,v)={P}_{C}\left({[}0,u]\times {[}1‒v,1])
(magenta).](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_001.jpg)
Values of the copula
There are many methods for constructing new copulas from given ones. One prominent and flexible class of such constructions are the so-called ordinal sums based on the Fréchet-Hoeffding bounds
Definition 2.5
If
are well-defined copulas. We shall call
Remark 2.6
Given a countable index set
2.1 Parameterized transformations of copulas
Based on ideas in [12,19,37,41,49,58,68], the following parameterized transformations have been introduced in [71, Definition 2.6] (compare also Remark 2.5 in that article) and studied extensively. Here, we present only a summary of the necessary notions and the most important properties related to these transformations.
Definition 2.7
Let
Note that, in the case of a copula
Formally, both transforms
Remark 2.8
If
we obtain in Figure 2, similarly as in Remark 2.4 and Figure 1, a geometric and stochastic interpretation of
![Figure 2
Values of the functions
D
1
{D}_{1}
and
E
1
{E}_{1}
defined by (2.9) at the black point
(
u
,
v
)
\left(u,v)
are given by the
P
C
{P}_{C}
-measures of the light blue and the blue rectangle (left), respectively, i.e.,
D
1
(
u
,
v
)
=
P
C
(
[
0
,
u
]
×
[
0
,
v
]
)
{D}_{1}\left(u,v)={P}_{C}\left(\left[0,u]\times \left[0,v])
(light blue) and
E
1
(
u
,
v
)
=
P
C
(
[
u
,
1
]
×
[
v
,
1
]
)
{E}_{1}\left(u,v)={P}_{C}\left({[}u,1]\times {[}v,1])
(blue). Similarly, the
P
C
{P}_{C}
-measures of the light red and the red rectangle (right) represent the values of the functions
D
2
{D}_{2}
and
E
2
{E}_{2}
at the black point
(
u
,
v
)
\left(u,v)
, i.e.,
D
2
(
u
,
v
)
=
P
C
(
[
0
,
u
]
×
[
v
,
1
]
)
{D}_{2}\left(u,v)={P}_{C}\left({[}0,u]\times {[}v,1])
(light red) and
E
2
(
u
,
v
)
=
P
C
(
[
u
,
1
]
×
[
0
,
v
]
)
{E}_{2}\left(u,v)={P}_{C}\left({[}u,1]\times {[}0,v])
(red).](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_002.jpg)
Values of the functions
Both the
Definition 2.9
For an arbitrary copula
Given a copula
The following list of properties of the parameter sets
Proposition 2.10
These are the main properties of the four parameter sets
For each copula
Let
For the three basic copulas
The
If a function
Here are some other properties of bivariate copulas
The properties PQD and NQD are preserved by our transforms and their truncations for non-negative and non-positive parameters
Lemma 2.11
Let
If C is a PQD copula and
If
2.2 Dependence parameters
The
In Section 5, we shall be concerned with several widely used dependence parameters of a copula
Definition 2.12
Given a bivariate copula
Given a copula
We shall investigate their dependence parameters and briefly refer to the results for the dependence parameters of
For the readers’ convenience, we display the results as differences between the values of the dependence parameter of the transformed and of the original copula considered as such stressing the influence of the transformation process on the dependence behavior of the copula involved.
Proposition 2.13
[52] Let
Proposition 2.14
Let
The formulas in Propositions 2.13 and 2.14 readily show that, with the exception of Kendall’s
3 M- and W-ordinal sums of transformed copulas
In [71, Section 3], we have discussed families of functions
As mentioned earlier in Section 2, we may, starting from a family
Proposition 3.1
Consider an arbitrary family
Proof
Use, for the sake of brevity, the abbreviation
Note that the right-hand side of (3.1) is an M-ordinal sum of a family of functions
Since the equality of the two functions in (3.1) holds for all
Fix an arbitrary index
Then, we have
Concerning the right-hand side of (3.1), we obtain for each
Now, (3.3) and (3.4) show that (3.1) holds for all
Corollary 3.2
Consider an arbitrary family
Proof
Since an ordinal sum of functions is a copula if and only if all its summand functions are copulas and due to the equality of the functions
From Proposition 2.10(i), we know that each
Corollary 3.3
Consider an M-ordinal sum
Proof
Corollary 3.2 tells us that
The following is a particularly interesting result presenting the close relationship between the non-negative part of the parameter set
Theorem 3.4
Let
Proof
If
If
Suppose
Obviously, on the set
and, on the other hand,
the latter being equivalent to
Since
Since
Remark 3.5
Theorem 3.4 provides an insight on the non-negative part of
i.e.,
From Proposition 2.10 and [71, Example 3.11], we know that
Here is an example of a family of copulas
Example 3.6
Consider the family of M-ordinal sums
For the set of all parameters
(3.9)For each
(3.10)
and, in the family of the
From Corollary 3.2, we know that
Note also that, using a simple computation, we can show that
To verify (3.9) for
If
Conversely, assume that
Indeed, for each
Now, let us use the just proven claim (i) to have a closer look at the family of truncated
and to identify the exact form of the copulas
Note that for
showing that
Next, consider some
Since
Finally, a simple computation verifies that indeed,
![Figure 3
Diagonal sections of the M-ordinal sum
D
0.8
=
M
-
(
⟨
0
,
0.8
,
W
⟩
)
{D}_{0.8}=M\hspace{0.1em}\text{-}\hspace{0.1em}(\langle 0,0.8,W\rangle )
(blue), of its
[
‒
3
]
\left[‒3]
-transform
(
D
0.8
)
[
‒
3
]
{\left({D}_{0.8})}_{\left[‒3]}
(orange) and of the truncation of its
[
‒
3
]
\left[‒3]
-transformation
(
D
0.8
)
[
‒
3
]
¯
\overline{\hspace{-1em}{\left({D}_{0.8})}_{\left[‒3]}}
(magenta) as considered in Example 3.6. The slope of the gray dotted line segment equals 2.](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_003.jpg)
Diagonal sections of the M-ordinal sum
Coming back to arbitrary M-ordinal sum of copulas, it is straightforward that the
and vice versa.
Using (3.13) and the commutativity (2.3) with regard to the functions
Proposition 3.7
Consider an arbitrary family
Proof
For an arbitrary family
showing that (3.14) holds.□
In complete analogy to the previous results on M-ordinal sums, we can verify the following assertions:
Corollary 3.8
Let
The function
If, for each
Remark 3.9
If
We have, so far, discussed the
We will now turn to the
Consider an arbitrary family
Note that whenever
Following [71, Corollary 3.5 and Remark 3.6], we immediately obtain the following necessary condition for
Corollary 3.10
Consider an arbitrary family
In particular, for
Note that the case
In complete analogy and by following [71, Corollary 3.12] and [71, Remark 3.13] we can state the following analogous result for W-ordinal sums.
Corollary 3.11
Consider an arbitrary family
In particular, for
Example 3.12
For each pair
Due to Corollary 3.10 and Proposition 2.10(i), we know that
If
Structure of the M-ordinal sum
Example 3.13
(Continuation of Example 3.6) Consider again the family of M-ordinal sums
For each
In particular,
For
From
If the summand
Example 3.14
For each
Fix an
and coincides with
For
4
Π
-ordinal sums of transformed copulas
Having discussed M- and W-ordinal sums of transformed copulas in Section 3, we now turn to
When considering a
Therefore, in this section, we will work, for
of
Starting with a
As a consequence, in order to show that
To formalize our first results of this section (in particular, those in Lemma 4.1 and in Propositions 4.2 and 4.3), let us introduce for a
Lemma 4.1
Let
The function
The function
Recall the sets
Proposition 4.2
Let
we obtain for the parameter sets
Let us now investigate the function
Proposition 4.3
Let
If
(4.3)If
(4.4)If
Proof
In the case
i.e., (i)–(iii) hold for
If
for all
If
Using basic calculations, for each
Since
is non-negative, i.e., if and only if
(4.5)In particular, this means that, for each rectangle
In order to verify that the function
Fix a rectangle
(4.6)which must hold for all
(4.7)In order to check that this inequality holds for all
Case 1(ii).
In this case, condition (4.7) is equivalent to
Case 2(ii).
Now, (4.7) is equivalent to
Case 3(ii).
Because of
(iii) If
In analogy to the proof of (ii), we look at the functions
The absolute continuity of
is non-negative, i.e., if and only if
Finally, consider an arbitrary rectangle
Because of
(4.8)If
Case 1(iii).
In this case, (4.8) is equivalent to
Case 2(iii).
From
(4.9)For
Summarizing Cases 1(iii) and 2(iii), we obtain
Using the
Corollary 4.4
Let
If
If
If
Proof
If the
Now, using (2.3) and
In Table 1, the results of Proposition 4.3 and Corollary 4.4 are summarized. To be precise, the respective sets of parameters
Overview of the results of Proposition 4.3 and Corollary 4.4
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If
In Proposition 4.3 and Corollary 4.4, we have considered
Theorem 4.5
Assume
If
Proof of Theorem 4.5(ii) and (iii)
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|---|---|---|
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Proof
Since
If
For assertions (ii) and (iii), we now assume that
If
which yields
Using analogous arguments for the cases
Remark 4.6
Since we have always checked the 2-increasingness individually in each vertical stripe, some of the results of Lemma 4.1, Propositions 4.2 and 4.3, Corollary 4.4 as well as Theorem 4.5 can be generalized for special
Consider, for example, Propositions 4.2 and 4.3 and suppose that 0 is an accumulation point of the sequence of boundary points (i.e., there is no “first” vertical stripe and the considered
As an immediate consequence of Proposition 4.3 and Corollary 4.4 in combination with Theorem 4.5, we obtain examples of
Example 4.7
For an arbitrary
The sets
However, if one takes into account also the truncations by means of
For the rest of this section, we, therefore, restrict ourselves not only to the special
We shall investigate the impact of the truncation by
Consider the following three subsets of the unit square
Then, the
and for each
Recall the family of copulas
Then, for each
For each
The different colors and intensities of gray in Figure 6 characterize the formulas for the computation of the function
![Figure 6
Structure of the function
⟨
Π
,
M
⟩
[
θ
]
¯
\overline{{\langle \Pi ,M\rangle }_{\left[\theta ]}\hspace{0em}}
given by (4.12) for the parameters
θ
∈
]
1
,
2
]
\theta \in ]1,2]
(left),
θ
∈
]
2
,
4
]
\theta \in ]2,4]
(center), and
θ
∈
]
4
,
∞
[
\theta \in ]4,\infty {[}
(right).](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_006.jpg)
Structure of the function
In particular, for
Now, we are in the position to determine the parameters
![Figure 7
Contour plots of the copulas
⟨
Π
,
M
⟩
[
3
2
]
¯
\overline{{\langle \Pi ,M\rangle }_{\left[\tfrac{3}{2}]}\hspace{.01em}}
,
⟨
Π
,
M
⟩
[
3
]
¯
\overline{{\langle \Pi ,M\rangle }_{\left[3]}\hspace{.01em}}
, and
⟨
Π
,
M
⟩
[
5
]
¯
\overline{{\langle \Pi ,M\rangle }_{\left[5]}\hspace{.01em}}
(left to right) as introduced in (4.14).](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_007.jpg)
Contour plots of the copulas
Proposition 4.8
The function
Proof
For each
Now, the inequalities
Turning to positive values of
Case 1.
Case 2.
From (4.14), it follows that
Case 3.
From
Case 4.
From
Case 5.
Taking into account that
Now, note that the parameter range
In summary, the function
The family of copulas
5 Special families of copulas and their dependence parameters
Finally, we present some examples of distinguished families of copulas, which have been investigated in Sections 3 and 4 of this manuscript or (as in the case of Examples 5.2 or 5.4) in one of our related articles. We also deal with the dependence parameters mentioned in Definition 2.12, i.e., with Spearman’s
Not all the families mentioned in this article are comprehensive (even if they have the Fréchet-Hoeffding bounds
![Figure 8
Dependence parameters of the ordinal sum
(
D
ε
)
ε
∈
[
0
,
1
]
{\left({D}_{\varepsilon })}_{\varepsilon \in {[}0,1]}
in Examples 3.6 and 5.1 (compare Table 3). Observe that each graph of the dependence parameters crosses the horizontal axis at a different value of
ε
\varepsilon
. Zooming into the blue rectangle
[
0.68
,
0.86
]
×
[
‒
0.2
,
0.2
]
{[}0.68,0.86]\times {[}‒0.2,0.2]
, the roots
ε
τ
{\varepsilon }_{\tau }
,
ε
β
{\varepsilon }_{\beta }
,
ε
ρ
{\varepsilon }_{\rho }
,
ε
γ
{\varepsilon }_{\gamma }
, and
ε
ϕ
{\varepsilon }_{\phi }
(whose exact values are given in (5.1)) can be easily distinguished (right). Note that these roots also appear in Figure 9.](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_008.jpg)
Dependence parameters of the ordinal sum
![Figure 9
Regions with positive (light blue) and negative (light yellow) values of the dependence parameters for the family of copulas
(
(
D
ε
)
[
θ
]
)
ε
∈
]
0
,
1
[
,
θ
∈
0
,
1
ε
{({\left({D}_{\varepsilon })}_{\left[\theta ]})}_{\varepsilon \in ]0,1{[},\theta \in \left[0,\tfrac{1}{\varepsilon }\right]}
in Examples 3.6 and 5.1 (compare Table 3). Points
(
ε
,
θ
)
\left(\varepsilon ,\theta )
with
ε
∈
[
0
,
1
]
\varepsilon \in \left[0,1]
and
θ
∈
1
,
1
ε
\theta \in \left[1,\frac{1}{\varepsilon }\right]
also belong to the light blue region with positive values of the respective dependence parameter. Each curve between the light blue and the light yellow region visualizes the set of points
(
ε
,
θ
)
\left(\varepsilon ,\theta )
for which the respective dependence parameter vanishes. For the values
ε
ϱ
{\varepsilon }_{\varrho }
,
ε
τ
{\varepsilon }_{\tau }
,
ε
β
{\varepsilon }_{\beta }
,
ε
γ
{\varepsilon }_{\gamma }
, and
ε
ϕ
{\varepsilon }_{\phi }
see (5.1) and Figure 8.](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_009.jpg)
Regions with positive (light blue) and negative (light yellow) values of the dependence parameters for the family of copulas
Different combinations of families of copulas in the last three examples lead to special types of comprehensive families. Indispensable tools for gaining deeper insights into these families of copulas are our five dependence parameters: Spearman’s
In Example 5.2, truncations of the
The combination of two rather simple
Finally, for copulas
Dependence parameters for the family of M-ordinal sums
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|||
| Spearman’s
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| Kendall’s
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| Blomqvist’s
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| Gini’s
|
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| Spearman’s footrule
|
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|
Example 5.1
(Continuation of Example 3.6) Recall the three families of ordinal sums and their transformations which are also copulas, as shown in (3.9) (see Example 3.6, where also the explicit formulas of these families of copulas were given).
The M-ordinal sums
The dependence parameters, i.e., Spearman’s
If, in the family of ordinal sums
Nevertheless, for each dependence parameter considered in this article there is exactly one real number
Looking now at the
showing that, by taking advantage of the
Since the subfamily
The following comprehensive family of copulas is obtained using
Example 5.2
In [71], the family of
or, in a more detailed manner,
implying that the family
The dependence parameters of the family
Dependence parameters of the family of copulas
Dependence parameters of the family of copulas
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|---|---|---|---|---|---|
| Range of
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1 |
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1 |
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1 |
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1 |
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1 |
Now, we combine a family of
Example 5.3
Consider the following combination
Obviously,
![Figure 11
Going from countermonotonicity through independence to comonotonicity, using the family of copulas
(
C
ε
)
ε
∈
]
−
1
,
1
[
{({C}_{\varepsilon })}_{\varepsilon \in ]-1,1{[}}
considered in (5.4) in Example 5.3 (observe that Spearman’s
ϱ
\varrho
and Kendall’s
τ
\tau
coincide whenever
ε
≥
0
\varepsilon \ge 0
).](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_011.jpg)
Going from countermonotonicity through independence to comonotonicity, using the family of copulas
Dependence parameters for the family of W-ordinal sums
| Ordinal sums |
|
|
|---|---|---|
| Spearman’s
|
|
|
| Kendall’s
|
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|
| Blomqvist’s
|
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| Gini’s
|
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| Spearman’s footrule
|
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|
Another combination of two families of copulas was considered in [71, Section 4], starting with some copula
Example 5.4
Recall the copulas
The set of all parameters
Looking at the special case that
The values
We can use the functions
Observe that
Dependence parameters for
| Copula |
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|---|---|---|---|
| range of
|
]‒∞
|
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| Spearman’s
|
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| Kendall’s
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| Blomqvist’s
|
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| Gini’s
|
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|
| Spearman’s footrule
|
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|
![Figure 12
Contour plots of several members of the family
(
J
θ
)
θ
∈
[
−
1
,
1
]
{({J}_{\theta })}_{\theta \in {[}-1,1]}
(see (5.5) in Example 5.4):
J
−
1
=
W
{J}_{-1}=W
,
J
−
0.4
{J}_{-0.4}
,
J
0
=
Π
{J}_{0}=\Pi
,
J
0.6
{J}_{0.6}
, and
J
1
=
M
{J}_{1}=M
(from left).](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_012.jpg)
Contour plots of several members of the family
![Figure 13
Dependence parameters of the family
(
J
θ
)
θ
∈
[
−
1
,
1
]
{({J}_{\theta })}_{\theta \in {[}-1,1]}
as considered in Example 5.4 (5.5). Note that three out of five dependence parameters coincide for
θ
<
0
\theta \lt 0
, and four out of five coincide for
θ
≥
0
\theta \ge 0
.](/document/doi/10.1515/demo-2024-0007/asset/graphic/j_demo-2024-0007_fig_013.jpg)
Dependence parameters of the family
6 Concluding remarks
Countermonotonicity, independence, and comonotonicity may be seen as extreme cases of dependence behavior in stochastics. Building dependence complete families of copulas may provide a tool for identifying copulas with a prescribed dependence parameter value at hand. A possible approach for doing so is to start from one of the three basic copulas, i.e., allowing for countermonotonicity, independence, and comonotonicity as basic behavior, and performing “local” transformations as in case of ordinal sums, perturbing the basic behavior by adding parameterized functions related to the starting copula and performing truncation with the Fréchet-Hoeffding bounds to insure basic copula properties. Our results and examples have illustrated how flexible, but also how restrictive, the conditions for obtaining new copulas in some cases are and how different the development of the dependence parameters may be when considering such constructed (dependence complete and/or comprehensive) families of copulas.
Acknowledgement
All authors would like to thank the three anonymous reviewers whose detailed comments helped them to substantially improve the original submission.
-
Funding information: This work was supported by Johannes Kepler University Open Access Publishing Fund and the Federal State Upper Austria. Anna Kolesárová and Adam Šeliga gratefully acknowledged the support by the Project VEGA 1/0036/23.
-
Author contributions: Susanne Saminger-Platz: Concept, Methodology, Writing – original draft, Writing – review & editing; Funding acquisition, Examples. Anna Kolesárová: Writing – review & editing, Funding acquisition. Adam Šeliga: Writing – review & editing, Funding acquisition. Radko Mesiar: Concept, Methodology, Examples. Erich Peter Klement: Concept, Methodology, Writing – original draft, Writing – review & editing, Examples.
-
Conflict of interest: The authors declare that there is no conflict of interest.
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