Negative Dependence Concept in Copulas and the Marginal Free Herd Behavior Index

# Negative Dependence Concept in Copulas and the Marginal Free Herd Behavior Index

Jae Youn Ahn Department of Statistics, Ewha Womans University, 11-1 Daehyun-Dong, Seodaemun-Gu, Seoul 120-750, Korea.
###### Abstract

We provide a set of copulas that can be interpreted as having the negative extreme dependence. This set of copulas is interesting because it coincides with countermonotonic copula for a bivariate case, and more importantly, is shown to be minimal in concordance ordering in the sense that no copula exists which is strictly smaller than the given copula outside the proposed copula set. Admitting the absence of the minimum copula in multivariate dimensions greater than 2, the study of the set of minimal copulas can be important in the investigation of various optimization problems. To demonstrate the importance of the proposed copula set, we provide the variance minimization problem of the aggregated sum with arbitrarily given uniform marginals. As a financial/actuarial application of these copulas, we define a new herd behavior index using weighted Spearman’s rho, and determine the sharp lower bound of the index using the proposed set of copulas.

## 1 Introduction

The study of the dependence structure between random variables via copula is a classical problem in statistics and other applications. The ease of application of copulas has led to their popularity in various areas such as finance, insurance, hydrology and medical studies; see for example, Frees and Valdez (1998), Genest et al. (2007) and Cui and Sun (2004). This paper examines the mathematical property of copulas by focusing on their lower bound.

Every copula is bounded by Fréchet-Hoeffding lower and upper bounds. While Fréchet-Hoeffding upper bound corresponds to the maximum copula, Fréchet-Hoeffding lower bound is generally not a copula. Further, the minimum copula does not exist in general in high dimensions greater than ; see, for example, Kotz and Seeger (1992) and Joe (1997).

In the insurance and finance field, the maximum copula corresponds to the concept called comonotonicity (Dhaene et al., 2002b). In the respect of risk management, comonotonicity is an important concept, because it can be used to describe the perfect positive dependence between competing risks. Importantly it provides the solution to various optimization (maximization) problems. However, unlike the perfect positive dependence, mainly due to the absence of the minimum copula, controversy has remained even in the definition of negative extreme dependence In spite of these difficulties, the need for the concept of negative extreme dependence has remained in insurance and other applications because it may lead to solutions for related optimization problems. Many studies have investigated the negative extreme dependence in various contexts. Dhaene and Denuit (1999), Cheung and Lo (2014) and Cheung et al. (2015) defined the concept of mutual exclusivity which can be regarded as pairwise countermonotonic movements. On the other hand, (Wang and Wang, 2011) proposed the concept of complete mixability, which can be used to minimize the variance of the sum of random variables with given marginal distributions. Many papers have recently been published in this field (Puccetti et al., 2012; Puccetti and Wang, 2014; Wang and Wang, 2014; Bernard et al., 2014). While the concepts of mutual exclusivity and complete mixability are both useful in various fields of optimization problems, since their concepts both depend on the marginal distributions and are problem specific, they may not provide the general concept of negative dependence.

Lee and Ahn (2014b) proposed a set of negative dependence joint distributions, which is named as -countermonotonic copulas (-CM). The definition of -CM is known to be the definition of copula only. Furthermore, the set of -CM copulas is minimal in terms of concordance ordering: there is no copula which is strictly smaller in concordance ordering than the given -CM copula except -CM copulas. Admitting the absence of the minimum element in multivariate dimensions , the set of minimal copulas can be important in optimization problems. However, without understanding the further properties of -CM copulas, choosing the proper -CM copulas for the given optimization problem can be difficult. Furthermore, as specified in Puccetti and Wang (2014), -CM can be too general to be used for the negative extreme dependence For example, any vector with being a uniform[0,1] random variable is -CM, while it is close to a comonotonic random vector except the last element. Hence in this paper, to remove such an almost comonotonic case and emphasize the negative extreme dependence concept, we consider only a special subset of -CM copulas, which will be parameterized by the vector , where is a -dimensional positive Euclidean space. Such set of copulas will be named as -countermonotonic copulas (-CM). Due to the minimality property of the set of -CM copulas, which is inherited from -CM, we expect that the set of -CM copulas might be also useful in various optimization problems.

However, before we discuss the usefulness of -CM copulas in optimization problems, the existence of -CM copulas should be first investigated. While the existence of -CM copulas with

 →w=(1,⋯,1)∈Rd+

is well known in the literature, see, for example, Lee and Ahn (2014b), existence of -CM copulas is not guaranteed for general . This paper provides the equivalence condition for the existence of -CM copulas. For the proof and construction of the copula, we use a simple geometrical method to construct the copula. A similar result obtained by using an algebraic method can be found in a recent working paper by Wang and Wang (2014).

Since -CM is the property of the copula only, the usefulness of -CM may be limited to some optimization problems which do not depend on marginal distributions. Puccetti and Wang (2014) also note the possible limitedness of -CM (hence -CM) in solving optimization problems by commenting that any dependence concept which does not take into account marginal distributions may fail to solve optimization problems which depend on marginal distributions. Variance minimization of the aggregated sum with given marginal distributions, which is formally stated in (19) below, is one such example; detailed literature can be found in Gaffke and Rüschendorf (1981); Rüschendorf and Uckelmann (2002); Wang and Wang (2011); Puccetti and Wang (2014). As can be intuitively expected, and as will be shown in Section 5 below, it can be shown that no single copula universally minimizes the variance of the aggregated sum with arbitrarily given marginals. However, we will show that using a set of -CM copulas rather than a single copula can minimize the variance of the aggregated sum for varying marginal distributions when restricted to the uniform distribution family. While our result provides a general solution with no restriction on , a partial solution can be observed in Wang and Wang (2014) for some special cases of where they are mainly interested in so called joint mixability which aims for the constant aggregated sum. More detailed results will be provided in Section 5.

For a financial application of -CM, we provide a new definition of the herd behavior index. Herd behaviors describe the comovement of members in a group. Since herd behaviors in the stock markets are observed usually during financial crises (Dhaene et al., 2012; Choi et al., 2013), measuring the herd behavior can be important in managing financial risks. Focusing on the fact that the perfect herd behavior can be modeled with the comonotonicity, some herd behavior indices that measure the degree of comonotonicity via the concept of the (co)variance have been proposed (Dhaene et al., 2012, 2014a; Choi et al., 2013). Measuring the herd behavior using such herd behavior indices can be important as it has been shown to be an indicator of the market fear. However, while the concept of comonotonicity is free of marginal distribution (and hence so is the herd behavior), these herd behavior measures can depend on marginal distributions, as will be shown in Example 2 below. Alternatively, we define the new herd behavior index based on a weighted average of bivariate Spearman’s rho. This new herd behavior index is not affected by the marginal distributions by definition and will be shown to preserve the concordance ordering. We also show that the maximum and minimum of the new herd behavior are closely related with comonotonicity and -CM.

The rest of this paper is organized as follows. We first summarize the study notations, and briefly explain basic copula theory and countermonotonicity theory in Section 2. The concept of -CM is introduced in Section 3, and the existence of -CM copula is demonstrated in Section 4. Section 5 applies the concept of -CM to variance minimization problems. The definition and minimization of the new herd behavior index are discussed in in Section 6, which is followed by the conclusions.

## 2 Notations and Preliminary Results

### 2.1 Conventions

Let be integers and denotes dimensional Euclidean space. Especially, let be dimensional positive Euclidean space. Further is denoted by . We use to denote -variate vectors: especially, lower case

 →x=(x1,x2,⋯,xd)

denotes constant vectors in and upper case

 →X=(X1,X2,⋯,Xd)

denotes -variate random vectors. More specifically

 →u:=(u1,⋯,ud)and→w:=(w1,⋯,wd)

will be used to denote constant vectors in and , respectively. Finally, use to denote a uniform random variable.

Unless specified, we assume be a -dimensional random vector having as its cumulative distribution function defined by

 H(→x)=P(X1≤x1,⋯,Xd≤xd)for→x∈Rd,

and the marginal distribution of is for and . Define to be the Fréchet space of -variate random vectors with marginal distribution . Hence, . Equivalently, we also denote . We use to denote the special case of Fréchet space, where all marginal distributions are uniform.

This paper assumes that marginals distributions are continuous. According to Sklar (1959), given , there exists a unique function satisfying

 H(→x)=C(F1(x1),⋯,Fd(xd)).

The function is called a copula, which is also a distribution function on . Further information on copulas can be found, for example, Cherubini et al. (2004) and Nelsen (2006).

Any satisfies

where

 W(→u):=max{u1+⋯+ud−(d−1),0}andM(→u):=min{u1,⋯,ud}, (1)

for . and in (1) are called the Fréchet-Hoeffding lower and Fréchet-Hoeffding upper bounds, respectively. Note that is a cumulative distribution of a -variate random vector while is not in general. Let be a survival distribution function defined as

 ¯¯¯¯¯H(→x):=P(X1>x1,⋯,Xd>xd)for→x∈Rd.

For , the concordance ordering is defined by

 H(→x)≤H∗(→x)and¯¯¯¯¯H(→x)≤¯¯¯¯¯¯¯H∗(→x)for all→x∈Rd.

Furthermore, define if

 H(→x)=H∗(→x)

for any . Equivalently, denote if , where the cumulative distribution function of is . Unless specified,

 →U:=(U1,⋯,Ud),−→U∗:=(U∗1,⋯,U∗d),and−−→U∗∗:=(U∗∗1,⋯,U∗∗d)

are -variate random vectors in having copula , and as cumulative distributions functions, respectively. For example,

 P(U1≤u1,⋯,Ud≤ud)=C(→u)

for .

It will be convenient to define the minimal and minimum copulas. For , we define -dimensional copula as a minimum(maximal) copula if the inequality

 C∗≻(≺)C

for any -dimensional copula . Similarly, for , define -dimensional copula as a minimal(maximal) copula if the inequality

 C∗≺(≻)C

for some -dimensional copula implies . Define the set of copulas to be minimal in set concordance ordering if any and with

 C∗≺C

implies

 C∗∈C.

By definition, is minimal in set concordance ordering if is empty. Clearly, the definition of minimality in set concordance ordering is a weaker concept than the definition of minimal copula. In the minimality of set concordance ordering, the quality of the minimality depends on the size of the set. For example, Fréchet space is minimal in set concordance ordering. On the other hand, if has a single element, the definition of the minimality in set concordance ordering coincides with the definition of the minimal copula.

### 2.2 Review of d-Countermonotonicity

Comonotonicity has gained popularity in actuarial science and finance. Conceptually, a random vector is comonotonic if all of its components move in the same direction. Comonotonicity is useful in several areas, such as the bound problems of an aggregate sum (Dhaene et al., 2006; Cheung and Vanduffel, 2013) and hedging problems (Cheung et al., 2011). Recently, comonotonicity has been used in describing the economic crisis (Dhaene et al., 2012, 2014b; Choi et al., 2013).

Countermonotonicity is the opposite concept to comonotonicity. Conceptually, in the bivariate case, a random vector is countermonotonic if two components move in the opposite directions. The following classical results summarize the equivalent conditions of countermonotonicity in bivariate dimensions.

###### Definition 1.

A set is countermonotonic(comonotonic) if the following inequality holds

 (x1−y1)(x2−y2)≤(≥)0for all→x,→y∈R2.

is called countermonotonic(comonotonic) if it has countermonotonic(comonotonic) support.

###### Theorem 1.

For a bivariate random vector , we have the following equivalent statements.

1. is countermonotonic

2. For any

 P(→X≤→x)=max{F1(x1)+F2(x2)−1,0} (2)
3. For Uniform() random variable , we have

 →Xd=(F−11(U1),F−12(1−U1)).

While the extension of comonotonicity into multivariate dimensions is straightforward, there is no obvious extension of countermonotonicity into multivariate dimensions . As discussed in Lee and Ahn (2014b), the difficulty of the extension of countermonotonicity arises partially due to the lack of minimum copula. In this paper, we provide a set of minimal copulas, which can be viewed as a natural extension of countermonotonicity in two dimension into multivariate dimensions.

## 3 Weighted Countermonotonicity

As an extension of countermonotonicity or negative extreme dependence in multivariate dimensions, Lee and Ahn (2014b) introduced the concept of -CM. While -CM copulas are theoretically interesting, the existence and construction of -CM copulas with certain parametric functions remain unknown, and it may therefore be hard to apply -CM copulas to various optimization problems. Furthermore, the concept of -CM may be too general to describe the negative dependence concept as briefly specified in Puccetti and Wang (2014), where the example of was given. Alternatively, Lee and Ahn (2014b) introduced the concept of strict -CM as a special case of -CM, which is useful in some minimization problems. However, because of the symmetricity of strict -CM, it cannot be used for non-symmetric optimization problems, as will be explained in Section 5. For completeness in the paper, we have summarized the definitions and properties of (strict) -CM in the Appendix.

In this section, we introduce a new class of extremal negative dependent copulas, which will be called -Countermonotonic (-CM) copulas and can be interpreted as a set of minimal copulas as shown in Corollary 1 below. Remark 1 addresses that the set of -CM copulas can be interpreted as generalized strict -CM, and further shows that -CM copulas are the subset of -CM copulas.

###### Definition 2.

A -variate random vector is -CM if

 P(d∑i=1wiFi(Xi)=∑di=1wi2)=1.

Equivalently, we say that is -CM if is -CM. Furthermore, when is -CM, we define as a shape vector of .

-CM can be regarded as multivariate extension of countermonotonicity into multivariate dimensions. First, assume that is countermonotonic. Then, since is a continuous random vector, Theorem 1. iii concludes that

 F1(X1)+F2(X2) d=F1∘F−11(U1)+F2∘F−12(1−U1)

which in turn implies is -CM for any . On the other hand, assume that is -CM with . Then by Definition 2, we have

 F1(X1)+F2(X2)=1

with probability , which in turn concludes that the support of is countermonotonic. So we can conclude that is countermonotonic if and only if is -CM with .

As can be expected from Definition 2, -CM is a property of copula only, and this is summarized in the following lemma. The proof is similar to that of Lemma 1 in Lee and Ahn (2014b). However, the result in the following lemma is more useful as it shows that the shape vector is invariant to marginal distributions.

###### Lemma 1.

Let and be random vectors from the distribution functions

 H=C(F1,⋯,Fd)andH∗=C(F∗1,⋯,F∗d),

respectively, where marginal distribution functions, , are possibly different from marginal distribution functions, . Then is -CM if and only if is -CM.

###### Proof.

Since two random vectors and have copula as the same distribution functions, we have

 P(d∑j=1w1Fi(Xi)=∑di=1wi2)=1, (3)

if and only if

 P(d∑j=1w1F∗i(X∗i)=∑di=1wi2)=1. (4)

Hence we conclude that is -CM if and only if is -CM. ∎

In the following definition, we provide the copula version of -CM. Note that, for the property of -CM, it is enough to study the copula version of -CM, because -CM is a property of copula only. Hence, throughout this paper, we will use the following definition as the definition of -CM.

###### Definition 3.

A -variate random vector is -CM if

 P(d∑i=1wiUi=∑di=1wi2)=1. (5)

Equivalently, we say that is -CM if is -CM. Here, is called as a shape vector of .

###### Remark 1.

Note that -CM is -CM with parameter functions

 fi(y)=ciFi(y)

for and , where

 ci:=2wid∑j=1wj.

Furthermore, since -CM coincides with strict -CM when , the set of strict -CM copula is the subset of -CM copulas. in the Appendix. For convenience, we summarize the definitions of -CM and strict -CM in Definition 6 and Definition 7 in the Appendix.

The following corollary explains that the set of -CM copulas can be regarded to have minimality in concordance ordering as a set. Since -CM is a special case of -CM as shown in Remark 1, the proof of the following corollary is immediate from Lee and Ahn (2014b). However, for completeness in the paper, we present the proof in the Appendix.

###### Corollary 1.

For given , let be the set of -CM: i.e. is defined as

 C:={C∈Fd∣∣ C is →w-CM}.

Then is minimal in set concordance ordering.

As briefly mentioned in Section 1, since there is no minimum copula available for , it is clear that minimal copulas will play a key role in various minimization problems. In this sense, Corollary 1 addresses an important property of -CM copulas: the set of -CM copulas achieves minimality in the sense that there are no copulas strictly smaller than the -CM copula other than -CM copulas. Hence, the concept of -CM can be useful in various minimization/maximization problems as will be explained in Section 5 below. For a discussion of the usage of -CM copulas, it is essential to check the existence of -CM copulas as will be shown in the next section.

## 4 Condition to Achieve the Weighted Countermonotonicity

Depending on the given marginal distributions, (5) may not be always achieved. For example, for , none of can achieve the condition in (5). In this section, we provide the equivalence condition of the weight for the existence of -CM copula. Note that a similar result can be found in a recent working paper by Wang and Wang (2014), which explains an algebraic way of constructing -CM copulas. We first define the set of weights where -CM copulas exists.

###### Notation 1.

Define the set of weights in -dimensions as follows

 W3:={(w1,w2,w3)∈R3+∣∣3∑i=1wi≥2max{w1,w2,w3}}.

Note that the set is equivalent with the set of the line lengths in triangles (including degenerate triangles).

###### Lemma 2.

For any , there exists -CM copulas.

###### Proof.

For convenience, define

 z1:=w2+w3−w12w2,z2:=w3+w1−w22w3andz3:=w1+w2−w32w1,for % some(w1,w2,w3)∈R3+. (6)

and denote if

 u1w1+u2w2+u3w3=w1+w2+w32.

Now, let us consider the following three points

 →p1:=(1,z1,0),→p2:=(0,1,z2)and→p3:=(z3,0,1),

and observe that the points satisfy for . Hence any point on the line that connects and is again in : i.e.

 t→pi+(1−t)→pj∈Φ(→w) (7)

for any and . Further, by the assumption , the following inequalities can be derived

 0≤zi≤1fori∈{1,2,3},

which in turn implies

 t→pi+(1−t)→pj∈[0,1]3 (8)

for any and . Note that the trace of (8) is triangular in with vertices lying on , and .

Now for the given triangle with vertices , and , we give positive weights , , to each edge , and such that weights are uniformly distributed on each edge. Here we assume that that the sum of weights is given as , so that the weights on the edges of the triangle define a random vector . In defining as the cumulative distribution function of the random vector , our goal is to show that there exist the weights which make a copula.

To show that is a copula, it is enough to show that is -increasing and that the marginals of are a uniform distribution (Nelsen, 2006). Since is defined by the nonnegative weights that are distributed on the edges of the triangular, it is obvious that is -increasing. Now, it remains to show that marginals of are a uniform distribution. Since weights are uniformly distributed on each edge, it is enough to check uniformity on each vertex of the triangle, which is equivalent to show

 P(X1≤z1)=z1,P(X2≤z2)=z2andP(X3≤z3)=z3. (9)

Each equation in (9) is equivalent with

 z1 =m12⋅0+m23z1+m31⋅1, (10) z2 =m12⋅1+m23⋅0+m31z2, z3 =m12z3+m23⋅1+m31⋅0,

respectively. The solution of (10) is

 m12=(1−z1)(1−z2)(1−z3)−(1−z1)(1−z2)+(1−z1)(1−z1)(1−z2)(1−z3)+1, (11) m23=(1−z1)(1−z2)(1−z3)−(1−z2)(1−z3)+(1−z2)(1−z1)(1−z2)(1−z3)+1, m31=(1−z1)(1−z2)(1−z3)−(1−z3)(1−z1)+(1−z3)(1−z1)(1−z2)(1−z3)+1.

While for general , a tedious but straightforward calculation shows that, with defined in (6), defined in (11) always satisfies

 m1,2+m2,3+m3,1=1.

Finally, (7) derives that is -CM.

While is some vector in , it is worth mentioning that the definition (6) is crucial to guarantee that the solution of (9) satisfies

 m1,2+m2,3+m3,1=1.

In other words, , which satisfies (9) may not satisfy

 m1,2+m2,3+m3,1=1

for and arbitrarily given that does not satisfy the condition (6). For example, for the arbitrarily given , the solution of (9) defined in (11) has

 m1,2+m2,3+m3,1>1.

The following lemma is an extension of Lemma 2 into multivariate dimensions .

###### Lemma 3.

For given , if there exist disjoint subsets , and of such that

 ⎛⎝∑wi∈Awi,∑wi∈Bwi,∑wi∈Cwi⎞⎠∈W3andA∪B∪C={w1,⋯,wd}, (12)

then there exists a random vector whose marginals are uniform[] and it satisfies

 d∑i=1wiUi=∑di=1wi2. (13)
###### Proof.

Let be random vectors with marginals being uniform[0,1]. Further, let

 V1:=∑i∈AUi,V2:=∑i∈BUiandV3:=∑i∈CUi.

Now the proofs are trivial if we set ’s in the same subset as being comonotonic i.e. and are comonotonic if either , or . ∎

The following lemma provides the equivalence condition of (12), which is more intuitive and easy to verify.

###### Lemma 4.

For the given weight , we have the following inequality

 max{w1,⋯,wd}≤d∑i=1wi−max{w1,⋯,wd} (14)

if and only if there exist disjoint subsets , and of satisfying (12).

###### Proof.

First observe that (12) implies (14) is trivial. Hence it remains to show (14) implies (12). Without loss of generality, let . For any integer , define

 w∗2:=∑i∈ZOwiandw∗3:=∑i∈ZEwi

where and . Then it is straightforward to show that

 w1+w∗2≥w∗3andw1+w∗3≥w∗2 (15)

Hence, along with (15), if we assume that satisfies (14), we can conclude , which in turn implies (12) with , and . ∎

So far in Lemma 3 and Lemma 4, we have provided sufficient conditions for the existence -CM copula. Then the natural question is to check whether they are also necessary conditions or not. The following corollary shows that the condition in (14) is also a necessary condition for the existence of -CM copulas.

###### Corollary 2.

For the given weight , there exists random vector -CM random vector if and only if satisfies

 max{w1,⋯,wd}≤d∑i=1wi−max{w1,⋯,wd}. (16)
###### Proof.

It is enough to show that -CM implies (16). First, consider a weight such that one weight, say , is greater than the sum of all other weights

 w1>d∑i=2wi. (17)

Then, it is obvious that there does not exist any random vector whose marginals are uniform[] and satisfies (5): this can be easily verified using the following variance comparison;

 Var(w1U1)>Var(d∑i=2wiUi).

Hence, we can conclude that there does not exist -CM copula under the condition (17), which concludes the claim.

###### Remark 2.

For any satisfying (16), the choice of -CM copula is not unique. For example, in the proof of Lemma 2, we show how to construct -CM copula for . On the other hand, the construction method in the proof of Lemma 2 and the following choices of defined as

 z∗1:=w2+w3−w12w3,z∗2:=w3+w1−w22w1andz∗3:=w1+w2−w32w2,

and three points defined as

 →p∗1:=(1,0,z1),→p∗2:=(z2,1,0)and→p∗3:=(0,z3,1),

will derive another choice of -CM copula with .

In the bivariate case, Corollary 2 concludes that being -CM implies that which coincides with the concept of countermonotonicity as we already mentioned in Section 3. The following example shows the numerical example of the construction of -CM copula using the logic in the proof of Lemma 2.

###### Example 1.

Let . Since

 5=w1≤w2+w3=4+3,

we know that and, by Corollary 2, there exists a -CM random vector . Using the techniques used in (2), we can construct -CM random vector having mass , and uniformly distributed on the each edge , and , respectively. Here

and

 m12=611,m23=311andm31=211 (18)

Hence, for example, we have

 C(1,0.25,1/3) =1/3∗m31 =233.

Finally, Figure 1 shows the support of random vector .

## 5 Application to the Variance Minimization Problem

Finding the maximum and the minimum of variance in the aggregated sum with given marginal distributions is the classical optimization problem

 Var(d∑i=1Xi),for given Xi∼Fi,i=1,⋯,d. (19)

First of all, the maximization of (19) is straightforward using the comonotonic random vectors. For the minimization problem with , the answer is trivial with countermonotonic random variables. Regarding general dimensions , minimization of (19) was solved for some cases of marginal distributions (Gaffke and Rüschendorf, 1981; Rüschendorf and Uckelmann, 2002; Wang and Wang, 2011; Puccetti and Wang, 2014). However, minimization of (19) is not easy in general for . The following remark, which can be easily derived from Theorem 2.7 of Dhaene et al. (2014b), states that variance minimization problems are related with concordance ordering, which may offer some hints in the minimization of (19).

###### Remark 3.

Let be distribution functions having finite variances. If

 −→X∗,→X∈Fd(F1,⋯,Fd)

with , then

 Var(d∑i=1X∗i)≤Var(d∑i=1Xi).

From the remark, it is clear that the minimization and maximization of (19) is related with a minimum and maximum copula. While maximization of (19) is related with the comonotonic copula, due to the absence of the minimum copula for , the minimization of (19) is related with the set of the minimal copulas. Of course, the choice of the proper set of minimal copulas depends on the marginal distributions. Among many other choices of the marginal distributions in (19), this paper considers the uniform marginal distributions as shown in the following definition, which may be the simplest versions of (19). The following assumption is useful to simplify the notation in several theorems in this section.

Assume that .

###### Definition 4.

For given , define

 m−(→w):=inf{Var(d∑i=1˜Ui)∣∣∣˜Uiis uniform[0,wi] random variables, i=1,⋯,d} (20)

and

 m+(→w):=sup{Var(d∑i=1˜Ui)∣∣∣˜Uiis uniform[0,wi] random variables, i=1,⋯,d}.

where .

Equivalently, and can be written as

 m−(→w)=inf{Var(d∑i=1wiUi)∣∣∣→U∈Fd}

and

 m+(→w)=sup{Var(d∑i=1wiUi)∣∣∣→U∈Fd}.

The upper bound

 m+(→w)=112(d∑i=1wi)2

is achieved if and only if is comonotonic (Kaas et al., 2002; Dhaene et al., 2002a, b). Regarding the lower bound, when

 2max{w1,⋯,wd}≤d∑i=1wi, (21)

Corollary 2 concludes that

 m−(→w)=0.

However for which does not satisfy (21), minimization is not straightforward.

For which does not satisfy (21), Theorem 2 below finds the explicit expression for . More importantly, we also show that the minimum is achieved with -CM copulas even though may not be the same as . Finally, Corollary 3 provides the complete solution for and for any given . Before we examine the main results, it is convenient to present the following lemma and notations.

###### Lemma 5.

Let satisfy Assumption 1 and

 2w1=d∑i=1wi. (22)

Then the following inequality holds

 cov[U1,d∑i=1wiUi]≥0, (23)

where the equality holds if and only if is -CM.

###### Proof.

We first prove the inequality (23) as

 cov[U1,d∑i=1wiUi] =d∑i=1cov[U1,wiUi] =w1Var(U1)+d∑i=2wicorr[U1,Ui]√Var(U1)√Var(Ui) ≥w1Var(U1)−d∑i=2wi√Var(U1)√Var(Ui) =0

where the inequality arises from the fact that correlation of any two random variables is greater than , and the last equality is from the condition (22). Furthermore, since satisfies the condition (16), Corollary 2 concludes that the inequality in (23) becomes equality if and only if is -CM. ∎

###### Notation 2.

For given , define as

 w∗i:=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩wi; if 2wi≤d∑j=1wjd∑j=1wi−wi; if 2wi>d∑j=1wj (24)

for . Then, one can easily confirm . Further, let

 l(→w) :=1